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 $\kappa$, the algorithm is shown to produce with an $\mathcal{O}\left({\kappa }^{5/4}{d}^{1/4}{\epsilon }^{-1/2}\right)$ complexity samples from a distribution that, in Wasserstein distance, is at most $\epsilon >0$ away from the target distribution.

arXiv comment29 pages, 2 figures

References in corpus (13)

• MCMC using Hamiltonian dynamics
• User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient
• Underdamped Langevin MCMC: A non-asymptotic analysis
• Geometric integrators and the Hamiltonian Monte Carlo method
• Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm
• High-Order Langevin Diffusion Yields an Accelerated MCMC Algorithm
• The Randomized Midpoint Method for Log-Concave Sampling
• Complexity of randomized algorithms for underdamped Langevin dynamics
• Order conditions for sampling the invariant measure of ergodic stochastic differential equations on manifolds
• Bounding the error of discretized Langevin algorithms for non-strongly log-concave targets
• Implicit Langevin Algorithms for Sampling From Log-concave Densities
• The shifted ODE method for underdamped Langevin MCMC
• The connections between Lyapunov functions for some optimization algorithms and differential equations