Predescu Cristian
Department of Chemistry and Kenneth S. Pitzer Center for Theoretical Chemistry, University of California, Berkeley, California 94720, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2007 Jul;76(1 Pt 2):016704. doi: 10.1103/PhysRevE.76.016704. Epub 2007 Jul 18.
The efficiency of Monte Carlo samplers is dictated not only by energetic effects, such as large barriers, but also by entropic effects that are due to the sheer volume that is sampled. The latter effects appear in the form of an entropic mismatch or divergence between the direct and reverse trial moves. We provide lower and upper bounds for the average acceptance probability in terms of the Rényi divergence of order 1/2 . We show that the asymptotic finitude of the entropic divergence is the necessary and sufficient condition for nonvanishing acceptance probabilities in the limit of large dimension. Furthermore, we demonstrate that the upper bound is reasonably tight by showing that the exponent is asymptotically exact for systems made up of a large number of independent and identically distributed subsystems. For the last statement, we provide an alternative proof that relies on the reformulation of the acceptance probability as a large deviation problem. The reformulation also leads to a class of low-variance estimators for strongly asymmetric distributions. We show that the entropy divergence causes a decay in the average displacements with the number of dimensions n that are simultaneously updated. For systems that have a well-defined thermodynamic limit, the decay is demonstrated to be n(-1/2) for random-walk Monte Carlo and n(-1/6) for smart Monte Carlo (SMC). Numerical simulations of the Lennard-Jones 38 (LJ(38)) cluster show that SMC is virtually as efficient as the Markov chain implementation of the Gibbs sampler, which is normally utilized for Lennard-Jones clusters. An application of the entropic inequalities to the parallel tempering method demonstrates that the number of replicas increases as the square root of the heat capacity of the system.
蒙特卡罗采样器的效率不仅取决于能量效应，如大的势垒，还取决于由于所采样的巨大体积而产生的熵效应。后一种效应以直接和反向试验移动之间的熵失配或散度的形式出现。我们根据1/2阶的Rényi散度给出了平均接受概率的上下界。我们表明，熵散度的渐近有限性是在大维度极限下接受概率不为零的充要条件。此外，我们通过证明对于由大量独立同分布子系统组成的系统，指数是渐近精确的，从而表明上界相当紧密。对于最后一个陈述，我们提供了另一种证明，该证明依赖于将接受概率重新表述为一个大偏差问题。这种重新表述还导致了一类针对强非对称分布的低方差估计器。我们表明，熵散度会导致平均位移随着同时更新的维度数n而衰减。对于具有明确热力学极限的系统，对于随机游走蒙特卡罗，衰减被证明为n^(-1/2)，对于智能蒙特卡罗（SMC），衰减为n^(-1/6)。对 Lennard-Jones 38（LJ(38)）团簇的数值模拟表明，SMC 实际上与通常用于 Lennard-Jones 团簇的吉布斯采样器的马尔可夫链实现一样高效。熵不等式在并行回火方法中的应用表明，副本数量随着系统热容量的平方根增加。
