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 κ\sqrt{d}/\varepsilon, where κ is the condition number, d is the dimension, and \varepsilon is the target precision: this improves by a factor of \sqrt{d}/\varepsilon over the previous state-of-the-art results.