Wasserstein mixing time of the unadjusted Langevin algorithm

We provide new estimates in Wasserstein distance for the asymptotic bias of the unadjusted Langevin algorithm, in the classical setting of log-smooth strongly log-concave measures. Our bound implies a Wasserstein mixing time of order $\kappa \sqrt{d}\mathrm{/}\epsilon$, where $\kappa$ is the condition number, $d$ is the dimension, and $\epsilon$ is the target precision: this improves by a factor of $\sqrt{d}\mathrm{/}\epsilon$ over the previous state-of-the-art results.