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 a Lipschitz gradient. We prove that, initialized with a Gaussian random vector that has sufficiently small variance, iterating the LMC algorithm for $\widetilde{\mathcal{O}}(\lambda^2 d\epsilon^{-1})$ steps is sufficient to reach $\epsilon$-neighborhood of the target in both Chi-squared and Renyi divergence, where $\lambda$ is the logarithmic Sobolev constant of $\nu_*$. Our results do not require warm-start to deal with the exponential dimension dependency in Chi-squared 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 both of these metrics, under the same assumptions. Translating this rate to other metrics, our results also recover the state-of-the-art rate estimates in KL divergence, total variation and $2$-Wasserstein distance in the same setup. Finally, as we rely on the logarithmic Sobolev inequality, our framework covers a range of non-convex potentials that are first-order smooth and 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 $\ nu ⁇ = e ⁇ - f} 进行抽样研究,当潜在美元满足强烈的分散性条件时,我们用利普施茨梯度平滑第一阶。我们证明,如果先用高斯随机矢量启动,其差异小到足够小的高斯随机矢量,再使用LMC 算法,以美元宽度= (Llambda2 d\ epsilon ⁇ -1} ),美元就足以达到奇斯和雷尼差异中目标的近距离值。我们证明,如果用LMC算法计算出,那么在基斯偏差值和雷尼的距离值上, 美元是平坦度的直径比值, 在先前的平流率中,我们所知道的平偏差值比值 。