Properties and limitations of geometric tempering for gradient flow dynamics

arXiv stat.ML / 4/23/2026

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Key Points

The paper frames sampling from a target distribution as an optimization problem over probability distributions using Kullback–Leibler divergence.
It studies how “geometric tempering” changes the behavior of Wasserstein and Fisher–Rao gradient flows when the target distribution is replaced by a time-varying sequence of moving targets.
The authors prove that, in continuous time, convergence occurs exponentially and provide new convergence bounds for both Wasserstein and Fisher–Rao settings.
They analyze common time discretizations and show that, for the Fisher–Rao gradient flow, geometric mixtures of the initial and target distributions never improve convergence speed (in either continuous or discrete time).
Finally, the work characterizes the gradient-flow structure of tempered dynamics and derives adaptive tempering schedules to improve practical performance.

Abstract

We consider the problem of sampling from a probability distribution

\pi

. 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

\pi

. We consider the effect of replacing

\pi

with a sequence of moving targets

(\pi_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.