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.

翻译：考虑从概率分布π中采样的问题。众所周知，该问题可表述为概率分布空间上的优化问题，其目标是最小化与π之间的Kullback--Leibler散度。本文研究了在Wasserstein和Fisher--Rao梯度流中，通过几何升温将π替换为移动目标序列$(π_t)_{t\ge0}$的影响。我们证明了在连续时间情形下指数收敛性，为两种情况提供了全新的界。此外，我们分析了常见的时间离散化方法及其收敛性质。研究表明，在Fisher--Rao情形下，将目标分布替换为初始分布与目标分布的几何混合，无论在连续时间还是离散时间中均不会加速收敛。最后，我们探讨了升温动力学的梯度流结构，并推导了新颖的自适应升温调度方案。