We establish an information complexity lower bound of randomized algorithms for simulating underdamped Langevin dynamics. More specifically, we prove that the worst $L^2$ strong error is of order $\Omega(\sqrt{d}\, N^{-3/2})$, for solving a family of $d$-dimensional underdamped Langevin dynamics, by any randomized algorithm with only $N$ queries to $\nabla U$, the driving Brownian motion and its weighted integration, respectively. The lower bound we establish matches the upper bound for the randomized midpoint method recently proposed by Shen and Lee [NIPS 2019], in terms of both parameters $N$ and $d$.

翻译：更具体地说,我们证明最差的2美元强烈错误是美元=Omega(\\sqrt{d ⁇,N ⁇ -3/2}),这是用任何随机算法解决一个以美元维度不足的Langevin动态组成的家庭。 任何随机算法都只询问美元=nabla U$、驱动布朗运动及其加权集成,而仅询问美元=nabla U$、驱动布朗运动及其加权集成。 更低的界限与Shen和Lee最近提议的随机中点方法的上限匹配,分别为$=NPS 2019和$。