Sampling algorithms based on discretizations of Stochastic Differential Equations (SDEs) compose a rich and popular subset of MCMC methods. This work provides a general framework for the non-asymptotic analysis of sampling error in 2-Wasserstein distance, which also leads to a bound of mixing time. The method applies to any consistent discretization of contractive SDEs. When applied to Langevin Monte Carlo algorithm, it establishes $tilde{mathcal{O}}left( frac{sqrt{d}}{epsilon} right)$ mixing time, 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 at infinity. This bound improves the best previously known $tilde{mathcal{O}}left( frac{d}{epsilon} right)$ 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.
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