Langevin Monte Carlo for strongly log-concave distributions: Randomized midpoint revisited
This work addresses the challenge of scalable sampling in high-dimensional settings for researchers in computational statistics and machine learning, representing an incremental improvement over existing methods.
The paper tackles the problem of sampling from strongly log-concave distributions by analyzing the randomized midpoint discretization for kinetic Langevin diffusion, resulting in a nonasymptotic and computable upper bound on the Wasserstein-2 error, with improved guarantees for Euler discretization that reduce dependence on condition numbers.
We revisit the problem of sampling from a target distribution that has a smooth strongly log-concave density everywhere in $\mathbb R^p$. In this context, if no additional density information is available, the randomized midpoint discretization for the kinetic Langevin diffusion is known to be the most scalable method in high dimensions with large condition numbers. Our main result is a nonasymptotic and easy to compute upper bound on the Wasserstein-2 error of this method. To provide a more thorough explanation of our method for establishing the computable upper bound, we conduct an analysis of the midpoint discretization for the vanilla Langevin process. This analysis helps to clarify the underlying principles and provides valuable insights that we use to establish an improved upper bound for the kinetic Langevin process with the midpoint discretization. Furthermore, by applying these techniques we establish new guarantees for the kinetic Langevin process with Euler discretization, which have a better dependence on the condition number than existing upper bounds.