Geometric tempering is a popular approach to sampling from challenging multi-modal probability distributions by instead sampling from a sequence of distributions which interpolate, using the geometric mean, between an easier proposal distribution and the target distribution. In this paper, we theoretically investigate the soundness of this approach when the sampling algorithm is Langevin dynamics, proving both upper and lower bounds. Our upper bounds are the first analysis in the literature under functional inequalities. They assert the convergence of tempered Langevin in continuous and discrete-time, and their minimization leads to closed-form optimal tempering schedules for some pairs of proposal and target distributions. Our lower bounds demonstrate a simple case where the geometric tempering takes exponential time, and further reveal that the geometric tempering can suffer from poor functional inequalities and slow convergence, even when the target distribution is well-conditioned. Overall, our results indicate that geometric tempering may not help, and can even be harmful for convergence.

翻译：几何退火是一种流行的采样方法，用于从具有挑战性的多模态概率分布中采样，其通过从一系列分布中采样来实现，这些分布使用几何平均在易于处理的提议分布与目标分布之间进行插值。本文从理论上研究了当采样算法采用朗之万动力学时该方法的合理性，并证明了上界与下界。我们的上界是在函数不等式条件下的首次文献分析。这些上界断言了连续时间与离散时间下退火朗之万动力学的收敛性，且其最小化过程为某些提议分布与目标分布对导出了闭式最优退火调度。我们的下界展示了一个简单案例，其中几何退火需要指数时间，并进一步揭示即使目标分布条件良好，几何退火仍可能受困于不良的函数不等式与缓慢收敛。总体而言，我们的结果表明几何退火可能无助于收敛，甚至可能对收敛产生不利影响。