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

TL;DR

This paper extends the Polyak-Lojasiewicz inequality to the space of probability measures and demonstrates its role in ensuring exponential convergence of mean-field birth-death processes for regularized optimization problems.

Contribution

It introduces a measure-space analogue of PLI and proves its applicability to analyze convergence of birth-death processes in mean-field optimization.

Findings

PLI holds for broad class of energy functions with KL-divergence regularization.

Establishes exponential convergence of birth-death processes under the measure-space PLI.

Provides theoretical foundation for convergence analysis in measure-based optimization algorithms.

Abstract

The Polyak-Lojasiewicz inequality (PLI) in $\mathbb{R}^d$ 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 $\mathcal{P}(\mathbb{R}^d)$ 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.