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arXiv · 2206.02774

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

Linshan Liu·Mateusz B. Majka·Łukasz Szpruch
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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.

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Linshan Liu, Mateusz B. Majka, Łukasz Szpruch. 2022-06-06. Polyak-\L ojasiewicz inequality on the space of measures and convergence of mean-field birth-death processes. https://doi.org/10.1007/s00245-022-09962-0

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