Langevin 动力学中几何退火的收敛性与局限性的可证明结果
Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamics
AI总结：
本文从理论上分析了基于 Langevin 动力学的几何退火采样方法，给出连续与离散时间下的收敛上界及闭式最优退火调度，并通过下界证明该方法在简单情形下可能需指数时间甚至损害收敛。
摘要：
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，其做法是改为从一系列分布中采样，这些分布利用几何平均在较容易的建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为 Langevin 动力学时该方法的合理性，并证明了上界和下界。我们的上界是文献中首个在函数不等式下给出的分析。它们证明了连续时间和离散时间下退火 Langevin 方法的收敛性，并且对这些上界进行最小化可以得到某些建议分布与目标分布对的闭式最优退火调度。我们的下界展示了一个简单情形，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，几何退火也可能受到较差的函数不等式和缓慢收敛的影响。总体而言，我们的结果表明几何退火可能没有帮助，甚至可能对收敛有害。
英文摘要：
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.
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