We prove bounds on the variance of a function $f$ under the empirical measure of the samples obtained by the Sequential Monte Carlo (SMC) algorithm, with time complexity depending on local rather than global Markov chain mixing dynamics. SMC is a Markov Chain Monte Carlo (MCMC) method, which starts by drawing $N$ particles from a known distribution, and then, through a sequence of distributions, re-weights and re-samples the particles, at each instance applying a Markov chain for smoothing. In principle, SMC tries to alleviate problems from multi-modality. However, most theoretical guarantees for SMC are obtained by assuming global mixing time bounds, which are only efficient in the uni-modal setting. We show that bounds can be obtained in the truly multi-modal setting, with mixing times that depend only on local MCMC dynamics.

翻译：我们证明了在序贯蒙特卡洛（SMC）算法所获样本的经验测度下，函数 $f$ 的方差界，其时间复杂度依赖于局部而非全局的马尔可夫链混合动态。SMC 是一种马尔可夫链蒙特卡洛（MCMC）方法，它首先从已知分布中抽取 $N$ 个粒子，然后通过一系列分布，对粒子进行重加权和重采样，并在每一步应用一个马尔可夫链进行平滑处理。从原理上讲，SMC 试图缓解多模态带来的问题。然而，SMC 的大多数理论保证都是通过假设全局混合时间界获得的，这仅在单模态设置下是高效的。我们证明了在真正的多模态设置下也可以获得界，其混合时间仅依赖于局部 MCMC 动态。