论文标题
双环未经调整的Langevin算法
Double-Loop Unadjusted Langevin Algorithm
论文作者
论文摘要
未经调整的langevin算法（ULA）是一种著名的一阶方法，用于对数符号概率分布进行采样。这项工作提出了ULA的新退火尺寸尺寸时间表，该时间表可以证明从光滑的对数孔分布中进行采样的新收敛保证，该分发未被现有的最新融合保证所涵盖。为了建立这一结果，我们得出了一个新的理论结合，将Wasserstein距离与任何两个对数符合分布之间的总变化距离联系起来，以补充Talagrand T2不平等的范围。此外，将此新的步长计划应用于现有的约束采样算法，我们显示了从约束的日志符号分布中采样的最新收敛率，以及改进的尺寸依赖性。
A well-known first-order method for sampling from log-concave probability distributions is the Unadjusted Langevin Algorithm (ULA). This work proposes a new annealing step-size schedule for ULA, which allows to prove new convergence guarantees for sampling from a smooth log-concave distribution, which are not covered by existing state-of-the-art convergence guarantees. To establish this result, we derive a new theoretical bound that relates the Wasserstein distance to total variation distance between any two log-concave distributions that complements the reach of Talagrand T2 inequality. Moreover, applying this new step size schedule to an existing constrained sampling algorithm, we show state-of-the-art convergence rates for sampling from a constrained log-concave distribution, as well as improved dimension dependence.
