The task of sampling from a high-dimensional distribution $\pi$ on $\R^d$ is a fundamental algorithmic problem with applications throughout statistics, engineering, and the sciences. Consider the Langevin diffusion on $\R^d$ \begin{ align*} \dif X_t=-\nabla U(X_t)dt+\sqrt{ 2}dB_t, \end{ align*} under mild conditions, it admits $\pi(\dif x)\propto \exp(-U(x))\dif x$ as its unique stationary distribution. Recently, Kandasamy and Nagaraj (2024) introduced a stochastic algorithm called Poisson Randomized Midpoint Langevin Monte Carlo (PRLMC) to enhance the rate of convergence towards the target distribution $\pi$. In this paper, we first show that under mild conditions, the PRLMC, as a Markov chain, admits a unique stationary distribution $\pi_\eta$ ($\eta$ is the step size) and obtain the convergence rate of PRLMC to $\pi_\eta$ in total variation distance. Then we establish a sharp error bound between $\pi_\eta$ and $\pi$ under the 2-Wasserstein distance. Finally, we propose a decreasing-step size version of PRLMC and provide its convergence rate to $\pi$ which is nearly optimal.
Categories: math.ST math.PR stat.TH