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

2025-07-06

AI Summary

This work investigates gradient-flow-based probabilistic sampling, focusing on which Bregman divergences—within the Bregman divergence family—enable efficient optimization using only the unnormalized density of a target distribution π, thereby bypassing the intractable normalization constant (i.e., partition function). Through rigorous theoretical analysis under canonical information-geometric metrics—including the Wasserstein, Fisher–Rao, and Hessian metrics—we establish that the Kullback–Leibler (KL) divergence is the unique Bregman divergence whose associated gradient flow depends solely on the unnormalized density of π, without requiring explicit knowledge of the partition function. This result unifies and characterizes the distinctive role of KL divergence from both information-geometric and optimal transport perspectives. It provides a principled theoretical foundation for designing normalization-free sampling algorithms, with direct implications for variational inference and non-equilibrium dynamical sampling methods.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Probabilistic Programming
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• 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 distribution in which we aim to minimise a divergence from $π$. and The optimisation problem is normally solved through gradient flows in the space of probability distribution 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 $π$.

Problem

• Sampling from probability distributions via optimization
• Minimizing divergences using gradient flow methods
• KL divergence uniqueness in avoiding normalizing constants

Innovation

• Uses Kullback-Leibler divergence for sampling
• Optimizes via gradient flows in probability space
• Avoids normalising constant knowledge requirement