Mean Square Error Analysis of Stochastic Runge--Kutta Integrators

We analyze the mean square error of stochastic Runge--Kutta integrators for overdamped Langevin dynamics whose potential is 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 and Wang, both of strong order $\frac32$ and weak order 2, evaluating 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.