Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations
This work addresses the challenge of efficient and accurate sampling from high-dimensional distributions for researchers in computational statistics and machine learning, offering incremental improvements in algorithm design.
The authors tackled the problem of quantifying the error between the invariant distribution of ergodic stochastic differential equations and their numerical approximations, focusing on strongly log-concave cases. They developed a framework to analyze various integrators and introduced a novel splitting method for underdamped Langevin dynamics, achieving a sample complexity of O(κ^{5/4} d^{1/4} ε^{-1/2}) to get within ε Wasserstein distance of the target distribution.
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 $\mathcal{O}\big(κ^{5/4} d^{1/4}ε^{-1/2} \big)$ complexity samples from a distribution that, in Wasserstein distance, is at most $ε>0$ away from the target distribution.