The Randomized Midpoint Method for Log-Concave Sampling
This improves sampling efficiency for statistics and machine learning applications, though it is incremental as it builds on existing MCMC methods.
The paper tackles the problem of sampling from log-concave distributions, proposing a new MCMC algorithm based on underdamped Langevin diffusion that achieves an error of ε·D in Õ(κ^{7/6}/ε^{1/3} + κ/ε^{2/3}) steps, significantly faster than the previous best of Õ(κ^{1.5}/ε) steps.
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 $ε\cdot D$ error (in 2-Wasserstein distance) in $\tilde{O}\left(κ^{7/6}/ε^{1/3}+κ/ε^{2/3}\right)$ steps, where $D\overset{\mathrm{def}}{=}\sqrt{\frac{d}{m}}$ is the effective diameter of the problem and $κ\overset{\mathrm{def}}{=}\frac{L}{m}$ is the condition number. Our algorithm performs significantly faster than the previously best known algorithm for solving this problem, which requires $\tilde{O}\left(κ^{1.5}/ε\right)$ steps. Moreover, our algorithm can be easily parallelized to require only $O(κ\log\frac{1}ε)$ 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.