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Properties and limitations of geometric tempering for gradient flow dynamics

arXiv 2604.20301 ↗First posted 22 Apr 2026stat.ML cs.LG stat.COcited by 0

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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.

Key facts. No citing papers in the map yet (data through 13 Oct 2026). Builds on 19 of its 29 references that are in the map. First posted on arXiv 22 Apr 2026.

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References · 19

  1. 2018Analysis of Langevin Monte Carlo via convex optimizationcited by 126
  2. 2024NETS: A Non-Equilibrium Transport Samplercited by 67
  3. 2019Accelerating Langevin Sampling with Birth-deathcited by 45
  4. 2023Learning Interpolations between Boltzmann Densitiescited by 42
  5. 2020An invitation to sequential Monte Carlo samplerscited by 33
  6. 2022Birth-death dynamics for sampling: Global convergence, approximations and their asymptoticscited by 30
  7. 2022Particle algorithms for maximum likelihood training of latent variable modelscited by 26
  8. 2024Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Samplingcited by 23
  9. 2023Sampling via Gradient Flows in the Space of Probability Measurescited by 20
  10. 2023An Explicit Expansion of the Kullback-Leibler Divergence along its Fisher-Rao Gradient Flowcited by 16
  11. 2023A connection between Tempering and Entropic Mirror Descentcited by 16
  12. 2024Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalitiescited by 13
7 more
  1. 2025Complexity Analysis of Normalizing Constant Estimation: from Jarzynski Equality to Annealed Importance Sampling and beyondcited by 11
  2. 2024Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamicscited by 10
  3. 2024Efficient, Multimodal, and Derivative-Free Bayesian Inference With Fisher-Rao Gradient Flowscited by 9
  4. 2023Adaptive Annealed Importance Sampling with Constant Rate Progresscited by 7
  5. 2025Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flowscited by 5
  6. 2025A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flowscited by 4
  7. 2025An operator splitting analysis of Wasserstein--Fisher--Rao gradient flowscited by 2

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