Optimal tuning of the hybrid Monte Carlo algorithm
Optimal tuning of the hybrid Monte Carlo algorithm
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DOI：
10.3150/12-bej414
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发表时间：
2013-11-01
期刊：
影响因子：
1.5
通讯作者：
Stuart, Andrew
中科院分区：
文献类型：
--
作者：
Beskos, Alexandros;Pillai, Natesh;Stuart, Andrew
研究了混合蒙特卡罗算法在高维空间中的性质。HMC开发了一个马尔可夫链可逆相对于一个给定的目标分布Pi使用可分离的哈密顿动力学与潜在的对数Pi。从玻尔兹曼分布中随机选择额外的动量变量，然后使用蛙跳方案离散连续时间哈密顿动力学。通过Metropolis-Hastings接受/拒绝规则消除诱导偏倚。在独立同分布分量的简化情形下，我们证明了当状态空间的维数d趋于无穷大时，为了获得O（1）的接受概率，蛙跳步长h应按h = l x d（-1/4）的比例缩放。因此，在高维中，HMC需要O（d（1/4））步来遍历状态空间。我们还确定分析的渐近最优接受概率，原来是0.651（到小数点后三位）。该值最佳地平衡了生成建议的成本，其随着l的增加而减少（因为需要更少的步骤来达到期望的最终集成时间），以及与获得接受所需的建议的平均数量相关的成本，其随着l的增加而增加。
We investigate the properties of the hybrid Monte Carlo algorithm (HMC) in high dimensions. HMC develops a Markov chain reversible with respect to a given target distribution Pi using separable Hamiltonian dynamics with potential -log Pi. The additional momentum variables are chosen at random from the Boltzmann distribution, and the continuous-time Hamiltonian dynamics are then discretised using the leapfrog scheme. The induced bias is removed via a Metropolis-Hastings accept/reject rule. In the simplified scenario of independent, identically distributed components, we prove that, to obtain an O(1) acceptance probability as the dimension d of the state space tends to infinity, the leapfrog step size h should be scaled as h = l x d(-1/4). Therefore, in high dimensions, HMC requires O(d(1/4)) steps to traverse the state space. We also identify analytically the asymptotically optimal acceptance probability, which turns out to be 0.651 (to three decimal places). This value optimally balances the cost of generating a proposal, which decreases as l increases (because fewer steps are required to reach the desired final integration time), against the cost related to the average number of proposals required to obtain acceptance, which increases as l increases.