Abstract

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 where 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).
It can achieve  error (in 2-Wasserstein distance) in steps, where is the effective diameter of the problem and is the condition number. Our algorithm performs significantly faster than the previ-
ously best known algorithm for solving this problem, which requires  steps [7, 15]. Moreover, our algorithm can be easily parallelized to require only parallel steps. To solve the sampling problem, we propose a new framework to discretize stochastic differential equations. We apply this framework to discretize and simulate ULD, which converges to the target distribution p* . The framework can be used to solve not only the log-concave sampling problem, but any problem that involves simulating (stochastic) differential equations.