# Properties and limitations of geometric tempering for gradient flow dynamics

arXiv: https://arxiv.org/abs/2604.20301 · HTML: https://mlsanity.com/paper/2604.20301/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 29 references that are in the map.
- First posted on arXiv 22 Apr 2026.

## Builds on (most cited first, 19 of 19)

- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [NETS: A Non-Equilibrium Transport Sampler](https://mlsanity.com/paper/2410.02711/) (2024)
- [Accelerating Langevin Sampling with Birth-death](https://mlsanity.com/paper/1905.09863/) (2019)
- [Learning Interpolations between Boltzmann Densities](https://mlsanity.com/paper/2301.07388/) (2023)
- [An invitation to sequential Monte Carlo samplers](https://mlsanity.com/paper/2007.11936/) (2020)
- [Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics](https://mlsanity.com/paper/2211.00450/) (2022)
- [Particle algorithms for maximum likelihood training of latent variable models](https://mlsanity.com/paper/2204.12965/) (2022)
- [Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling](https://mlsanity.com/paper/2407.16936/) (2024)
- [Sampling via Gradient Flows in the Space of Probability Measures](https://mlsanity.com/paper/2310.03597/) (2023)
- [An Explicit Expansion of the Kullback-Leibler Divergence along its Fisher-Rao Gradient Flow](https://mlsanity.com/paper/2302.12229/) (2023)
- [A connection between Tempering and Entropic Mirror Descent](https://mlsanity.com/paper/2310.11914/) (2023)
- [Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities](https://mlsanity.com/paper/2407.15693/) (2024)
- [Complexity Analysis of Normalizing Constant Estimation: from Jarzynski Equality to Annealed Importance Sampling and beyond](https://mlsanity.com/paper/2502.04575/) (2025)
- [Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamics](https://mlsanity.com/paper/2410.09697/) (2024)
- [Efficient, Multimodal, and Derivative-Free Bayesian Inference With Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2406.17263/) (2024)
- [Adaptive Annealed Importance Sampling with Constant Rate Progress](https://mlsanity.com/paper/2306.15283/) (2023)
- [Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2506.05905/) (2025)
- [A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flows](https://mlsanity.com/paper/2507.04330/) (2025)
- [An operator splitting analysis of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2511.18060/) (2025)

## Abstract (arXiv)

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 (πt)t≥0 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.