Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations

Abstract

We present a framework that allows for the non-asymptotic study of the 2-Wasserstein distance between the invariant distribution of an ergodic stochastic differential equation and the distribution of its numerical approximation in the strongly log-concave case. This allows us to study in a unified way a number of different integrators proposed in the literature for the overdamped and underdamped Langevin dynamics. In addition, we analyse a novel splitting method for the underdamped Langevin dynamics which only requires one gradient evaluation per time step. Under an additional smoothness assumption on a d-dimensional strongly log-concave distribution with condition number κ, the algorithm is shown to produce with an O κ 5/4 d 1/4 ǫ −1/2 complexity samples from a distribution that, in Wasserstein distance, is at most ǫ > 0 away from the target distribution.

Metropolis Adjusted Langevin Trajectories: a robust alternative to Hamiltonian Monte Carlo

“…For this reason, this splitting scheme has received a significant interest; see [13,15,56,64,70]. Several other splittings of the Langevin diffusion have been proposed and studied; see [1,69]. This work focuses on the OBABO update in order to preserve several properties of the Störmer-Verlet update, useful for constructing a Metropolis correction.…”

“…Unadjusted sampling approximations can be controlled by solving a trade-off between running the chain long enough to get close enough to the stationary measure, while choosing a time-step small enough in order to control the discretization error. Solving this tradeoff with respect to log-concave target densities Π has received a lot of attention lately; see [22,23,28,29,30,41] for the overdamped Langevin diffusion, [20,24,25,47,56,69] for the Langevin diffusion, and [8,12,14,19,48,49,50] for Hamiltonian dynamics. One limitation of unadjusted samplers is that whenever the discretization error scales polynomially with the time-step, the number of gradient evaluations required to reach a given precision will increase polynomially with the precision level at best.…”