论文标题
Langevin Monte Carlo在卡方和Renyi Divergence中的收敛
Convergence of Langevin Monte Carlo in Chi-Squared and Renyi Divergence
论文作者
论文摘要
我们使用未经调整的Langevin Monte Carlo（LMC）算法从目标分布$ν_* = E^{ - F} $研究采样，当潜在的$ f $满足强大的散发条件时，并且它具有lipschitz的梯度。我们证明，以$ \ widetilde {\ Mathcal {o}}（λ^2diε^{ - 1} $ to in to $ to $ to $ to $ to Chi $ i is的$ i is chi $ iviver，以$ \ mathcal {o}}（λ^2dε^{ - 1}）为$ i is $ is $ is chi $ iviver，以$ \ mathcal {o}}（λ^2dε^{ - 1}）为$ $ε$ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ - $ is reny， $ν_*$的对数Sobolev常数。我们的结果不需要温暖的启动来处理初始化时卡方差异的指数尺寸依赖性。特别是，对于强凸和一阶光滑电势，我们表明LMC算法实现了速率估计$ \ widetilde {\ Mathcal {o}}}}（dε^{ - 1}）$，在同一假设下，这两个指标中都提高了先前已知的速率。将此速率转换为其他指标，我们的结果还恢复了KL差异，总变化和$ 2 $ - WASSERSTEIN距离的最先进率估计值。最后，由于我们依靠对数Sobolev不等式，我们的框架涵盖了一系列非凸电势，它们是一阶光滑且在紧凑型区域之外表现出较强的凸度。
We study sampling from a target distribution $ν_* = e^{-f}$ 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 $\widetilde{\mathcal{O}}(λ^2 dε^{-1})$ steps is sufficient to reach $ε$-neighborhood of the target in both Chi-squared and Renyi divergence, where $λ$ is the logarithmic Sobolev constant of $ν_*$. 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 $\widetilde{\mathcal{O}}(dε^{-1})$ 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.
