Underdamped Langevin MCMC: A non-asymptotic analysis

TL;DR

This paper provides a non-asymptotic analysis of underdamped Langevin MCMC, showing it converges faster than overdamped Langevin MCMC under certain conditions, with a bound of (\u221a{d}/) steps for  error.

Contribution

It introduces a discretized underdamped Langevin MCMC algorithm with proven faster convergence rates than overdamped methods under smooth and strongly concave log-target distributions.

Findings

Achieves  error in (/) steps

Outperforms overdamped Langevin MCMC in convergence rate

Provides quantitative support for empirical advantages of HMC-like methods

Abstract

We study the underdamped Langevin diffusion when the log of the target distribution is smooth and strongly concave. We present a MCMC algorithm based on its discretization and show that it achieves $\varepsilon$ error (in 2-Wasserstein distance) in $\mathcal{O}(\sqrt{d}/\varepsilon)$ steps. This is a significant improvement over the best known rate for overdamped Langevin MCMC, which is $\mathcal{O}(d/\varepsilon^2)$ steps under the same smoothness/concavity assumptions. The underdamped Langevin MCMC scheme can be viewed as a version of Hamiltonian Monte Carlo (HMC) which has been observed to outperform overdamped Langevin MCMC methods in a number of application areas. We provide quantitative rates that support this empirical wisdom.