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* proportional to exp(-f(x)), where f : R-d -> 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 . D error (in 2-Wasserstein distance) in (O) over tilde(kappa(7/6)/epsilon(1/3) + kappa/epsilon(2/3)) steps, where D def = root d/m is the effective diameter of the problem and kappa def = L/m is the condition number. Our algorithm performs significantly faster than the previously best known algorithm for solving this problem, which requires (O) over tilde(kappa(1.5)/epsilon) steps [7, 15]. Moreover, our algorithm can be easily parallelized to require only O (kappa log 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.