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.

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