We study sampling from a target distribution $\nu_* = e^{-f}$ using the unadjusted Langevin Monte Carlo (LMC) algorithm when the potential $f$ satisfies a strong dissipativity condition and it is first-order smooth with Lipschitz gradient. We prove that, initialized with a Gaussian that has sufficiently small variance, $\widetilde{\mathcal{O}}(\lambda d\epsilon^{-1})$ steps of the LMC algorithm are sufficient to reach $\epsilon$-neighborhood of the target in Chi-square divergence, where $\lambda$ is the log-Sobolev constant of $\nu_*$. Our results do not require warm-start to deal with exponential dimension dependency in Chi-square divergence at initialization. In particular, for strongly convex and first-order smooth potentials, we show that the LMC algorithm achieves the rate estimate $\widetilde{\mathcal{O}}(d\epsilon^{-1})$ which improves the previously known rates in this metric, under the same assumptions. Translating to other metrics, our result also recovers the best-known rate estimates in KL divergence, total variation and $2$-Wasserstein distance in the same setup. Finally, as we rely on the log-Sobolev inequality, our framework covers a wide range of non-convex potentials that are first-order smooth and that exhibit strong convexity outside of a compact region.
翻译：我们用未调整的Langevin Monte Carlo (LMC) 算法从目标分布量 $\ nu = = e = = f} = = = = 美元进行抽样研究,如果潜在美元满足强烈的脱气条件,并且与Lipschitz 梯度相比是顺畅的。我们证明,如果先用一个具有足够小差异的Gaussian 开始, 美元全方位推算, == (lambda) = = e = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
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