The Randomized Midpoint Method for Log-Concave Sampling

Ruoqi Shen, Yin Tat Lee

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 $p^{*}\propto\exp(-f(x))$, 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 $D\defeq\sqrt{\frac{d}{m}}$ is the effective diameter of the problem and $\kappa\defeq\frac{L}{m}$ is the condition number. Our algorithm performs significantly faster than the previously best ULD based one, which needs $\tilde{O}\left(\kappa^{2}/\epsilon\right)$ steps [#cheng2017underdamped]. It also significantly outperforms the best known algorithm for solving this problem, which is based on Hamiltonian Monte Carlo (HMC) and requires $\tilde{O}\left(\kappa^{1.5}/\epsilon\right)$ steps [#chen2019optimal]. 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 applies 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 problems that involve simulating (stochastic) differential equations.

CMT Num: 1228