发表机构
Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文分析过阻尼Langevin动力学中随机Runge--Kutta积分器的均方误差，证明其分布以二阶Wasserstein距离收敛，误差阶最优，并在实验中验证。
AI 中文摘要
我们分析了用于过阻尼Langevin动力学的随机Runge--Kutta积分器的均方误差，该动力学的势能在有界区域外是凸的。通过一种分解，将局部误差拆分为一个均值为零的项和一个较小的余项，离散泊松方程将其矩转化为时间平均误差的界。我们对Yang和Wang提出的两个随机Runge--Kutta积分器进行了此分析，两者均具有强阶$\frac32$和弱阶2，仅需评估势能的梯度而不需要更高阶导数。我们证明，在Wasserstein-1距离下，它们的分布以二阶收敛到精确解的分布，至多相差一个对数因子，且一致地依赖于步数。对于具有有界三阶导数的测试函数，我们证明了步长为$h$的$N$步均方误差为$\mathcal O ( \frac{1}{Nh} + h^4 )$，这是离散化中的最优阶。实验测量了$\mathbb R^2$中非凸势能上的强阶和弱阶以及采样偏差，并在CIFAR-10的扩散模型上比较了这些积分器。
英文摘要
We analyze the mean square error of stochastic Runge-Kutta integrators for overdamped Langevin dynamics whose potential is strongly convex outside a bounded region. A decomposition splits the local error into a mean-zero term and a smaller remainder, and the discrete Poisson equation turns their moments into a bound on the error of a time average. We carry this out for two stochastic Runge-Kutta integrators proposed by Yang & Wang (2026), both of strong order $\frac32$ and weak order $2$, which evaluate the gradient of the potential and no higher derivative. We show that their laws approach the law of the exact solution at second order in Wasserstein-1 distance, up to a logarithm, uniformly in the number of steps. For a test function with bounded derivatives up to third order, we prove that the mean square error over $N$ steps with step size $h$ is $\mathcal O ( \frac{1}{Nh} + h^4 )$, which is the optimal order in the discretization. Experiments measure the strong and weak orders and the sampling bias on a nonconvex potential in $\mathbb R^2$, and compare the integrators on a diffusion model of CIFAR-10.
