# Polyak-L ojasiewicz inequality on the space of measures and convergence of mean-field birth-death processes

arXiv: https://arxiv.org/abs/2206.02774 · HTML: https://mlsanity.com/paper/2206.02774/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jun 2022.

## Built on by (most cited first, 6 of 6)

- [Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics](https://mlsanity.com/paper/2211.00450/) (2022)
- [Direct Distributional Optimization for Provable Alignment of Diffusion Models](https://mlsanity.com/paper/2502.02954/) (2025)
- [A Fisher-Rao gradient flow for entropic mean-field min-max games](https://mlsanity.com/paper/2405.15834/) (2024)
- [Non-convex entropic mean-field optimization via Best Response flow](https://mlsanity.com/paper/2505.22760/) (2025)
- [Global linear convergence of entropy-regularized softmax policy gradient beyond tabular MDPs](https://mlsanity.com/paper/2605.24939/) (2026)
- [Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization](https://mlsanity.com/paper/2606.27767/) (2026)

## Abstract (arXiv)

The Polyak-Lojasiewicz inequality (PLI) in ℝd is a natural condition for proving convergence of gradient descent algorithms. In the present paper, we study an analogue of PLI on the space of probability measures P(ℝd) and show that it is a natural condition for showing exponential convergence of a class of birth-death processes related to certain mean-field optimization problems. We verify PLI for a broad class of such problems for energy functions regularised by the KL-divergence.