  This article considers the popular MCMC method of unadjusted Langevin Monte
Carlo (LMC) and provides a non-asymptotic analysis of its sampling error in
2-Wasserstein distance. The proof is based on a refinement of mean-square
analysis in Li et al. (2019), and this refined framework automates the analysis
of a large class of sampling algorithms based on discretizations of contractive
SDEs. Using this framework, we establish an $\tilde{O}(\sqrt{d}/\epsilon)$
mixing time bound for LMC, without warm start, under the common log-smooth and
log-strongly-convex conditions, plus a growth condition on the 3rd-order
derivative of the potential of target measures. This bound improves the best
previously known $\tilde{O}(d/\epsilon)$ result and is optimal (in terms of
order) in both dimension $d$ and accuracy tolerance $\epsilon$ for target
measures satisfying the aforementioned assumptions. Our theoretical analysis is
further validated by numerical experiments.
