Mixing time guarantees for unadjusted Hamiltonian Monte Carlo
Mixing time guarantees for unadjusted Hamiltonian Monte Carlo
复制标题
DOI：
10.3150/21-bej1450
复制
发表时间：
2023-02-01
期刊：
影响因子：
1.5
通讯作者：
Eberle, Andreas
中科院分区：
文献类型：
--
作者：
Bou-rabee, Nawaf;Eberle, Andreas
我们提供了定量上限的总变差混合时间的马尔可夫链对应的未调整的哈密顿蒙特卡罗（uHMC）算法。对于两个一般类的模型和固定的时间离散化步长h，混合时间只依赖于几何上的尺寸。此外，我们提供了定量的上限之间的总变化距离的不变测度的uHMC链和真正的目标措施。因此，我们表明，目标分布（）μ的总变化距离的ε-精确的近似可以实现uHMC：（i）广泛的一类模型与O d3/4 <$-1/2 log（d/<$）（）梯度评估;和（ii）与O d1/2 <$-1/2 log（d/<$）梯度评估弱相互作用的平均场模型。证明是基于成功的uHMC实现上界的耦合建设。
We provide quantitative upper bounds on the total variation mixing time of the Markov chain corresponding to the unadjusted Hamiltonian Monte Carlo (uHMC) algorithm. For two general classes of models and fixed time discretization step size h, the mixing time is shown to depend only logarithmically on the dimension. Moreover, we provide quantitative upper bounds on the total variation distance between the invariant measure of the uHMC chain and the true target measure. As a consequence, we show that an epsilon-accurate approximation of the target distribution ( ) mu in total variation distance can be achieved by uHMC: (i) for a broad class of models with O d3/4 epsilon-1/2log(d/epsilon) ( ) gradient evaluations; and (ii) for mean field models with weak interactions with O d1/2 epsilon-1/2 log(d/epsilon) gradient evaluations. The proofs are based on the construction of successful couplings for uHMC that realize the upper bounds.