The Randomized Midpoint Method for Log-Concave Sampling

Sampling from log-concave distributions is a well researched problem that has many applications in statistics and machine learning. We study the distributions of the form p {*}\propto\exp(-f(x)), where f:\mathbb{R} {d}\rightarrow\mathbb{R} has an L -Lipschitz gradient and is m -strongly convex. In our paper, we propose a Markov chain Monte Carlo (MCMC) algorithm based on the underdamped Langevin diffusion (ULD). Our algorithm performs significantly faster than the previously best known algorithm for solving this problem, which requires \tilde{O}\left(\kappa {1.5}/\epsilon\right) steps \cite{chen2019optimal,dalalyan2018sampling}. Moreover, our algorithm can be easily parallelized to require only O(\kappa\log\frac{1}{\epsilon}) parallel steps.