Wasserstein distance estimates for the distributions of numerical approximations to ergodic stochastic differential equations
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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^1/4ϵ^-1/2) complexity samples from a distribution that, in Wasserstein distance, is at most ϵ>0 away from the target distribution.
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