Polyak-Łojasiewicz inequality on the space of measures and convergence of mean-field birth-death processes

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.

21 pages, revised version, accepted for publication in Applied Mathematics & Optimization. The final manuscript is available at Springer via https://doi.org/10.1007/s00245-022-09962-0

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