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). It can achieve $\epsilon\cdot D$ error (in 2-Wasserstein distance) in $\tilde{O}\left(\kappa^{{7/6}/\epsilon{1/3}+\kappa/\epsilon{2/3}\right)$} steps, where $D\overset{\mathrm{def}}{=}\sqrt{\frac{d}{m}}$ is the effective diameter of the problem and $\kappa\overset{\mathrm{def}}{=}\frac{L}{m}$ is the condition number. 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. Moreover, our algorithm can be easily parallelized to require only $O(\kappa\log\frac{1}{\epsilon})$ 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.