A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flows

Francesca Romana Crucinio

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arXiv · 2507.04330

A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flows

Francesca Romana Crucinio

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Abstract

We consider the problem of sampling from a probability distribution $\pi$ which admits a density w.r.t. a dominating measure. 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 a divergence from $\pi$. The optimisation problem is normally solved through gradient flows in the space of probability distributions with an appropriate metric. We show that the Kullback--Leibler divergence is the only divergence in the family of Bregman divergences whose gradient flow w.r.t. many popular metrics does not require knowledge of the normalising constant of $\pi$.

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Francesca Romana Crucinio. 2025-07-06. A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flows. https://arxiv.org/abs/2507.04330

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