Underdamped Langevin MCMC: A non-asymptotic analysis
This provides a theoretical foundation for faster sampling in Bayesian inference and machine learning, though it is incremental as it builds on existing Langevin and HMC methods.
The paper tackles the problem of sampling from smooth, strongly concave distributions by analyzing underdamped Langevin MCMC, showing it achieves ε error in O(√d/ε) steps, which improves over the O(d/ε²) steps of overdamped Langevin MCMC.
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.