通过几何退火优化采样速度，提升连续与离散时间下的收敛性。
Properties and limitations of geometric tempering for gradient flow dynamics
- 用几何退火构造动态目标分布，优化梯度流采样过程。
- 在连续时间下实现指数级收敛，给出Wasserstein与Fisher-Rao流的新界。
- 发现混合初始与目标分布无法加速收敛，适用于需稳定采样的研究者。
本文研究从概率分布 π 采样的问题，可转化为最小化相对于 π 的KL散度的优化问题。通过几何退火定义一系列移动目标分布 (π_t)_{t≥0}，分析其在Wasserstein与Fisher-Rao梯度流中的作用。结果表明，在连续时间内收敛呈指数级，并在两种情形下获得新边界。同时考察了常见的时间离散化方法，发现Fisher-Rao情形中，以初始与目标分布的几何混合替代原目标分布，无论连续或离散时间均无法提升收敛速度。此外，研究了退火动力学的梯度流结构，推导出新的自适应退火策略。
原文摘要 · Abstract (English)
We consider the problem of sampling from a probability distribution $π$. It is well known that this can be written as an optimisation problem over the space of probability distributions in which we aim to minimise the Kullback--Leibler divergence from $π$. We consider the effect of replacing $π$ with a sequence of moving targets $(π_t)_{t\ge0}$ defined via geometric tempering on the Wasserstein and Fisher--Rao gradient flows. We show that convergence occurs exponentially in continuous time, providing novel bounds in both cases. We also consider popular time discretisations and explore their convergence properties. We show that in the Fisher--Rao case, replacing the target distribution with a geometric mixture of initial and target distribution never leads to a convergence speed up both in continuous time and in discrete time. Finally, we explore the gradient flow structure of tempered dynamics and derive novel adaptive tempering schedules.
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