Properties and limitations of geometric tempering for gradient flow dynamics

2026-04-22

AI Summary

This work addresses the unclear convergence mechanism of geometric annealing strategies in efficient sampling from a target probability distribution. By embedding the geometric annealing schedule into both Wasserstein and Fisher–Rao gradient flows, the study systematically analyzes convergence behavior in both continuous and discrete time settings. The paper establishes novel exponential convergence bounds for these two classes of gradient flows for the first time, revealing an intrinsic limitation of geometric mixing within the Fisher–Rao framework—it cannot accelerate convergence. Building on this insight, the authors propose an adaptive annealing schedule with rigorous theoretical guarantees, offering a principled approach to improve sampling efficiency while respecting the fundamental constraints of the underlying geometry.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

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.

Problem

• geometric tempering
• gradient flow
• sampling
• convergence
• probability distribution

Innovation

• geometric tempering
• gradient flow
• convergence bounds
• adaptive tempering
• Fisher–Rao metric