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

Autor:	Liu, Linshan, Majka, Mateusz B., Szpruch, Łukasz
Rok vydání:	2022
Předmět:	Mathematics - Optimization and Control Mathematics - Probability
Druh dokumentu:	Working Paper
DOI:	10.1007/s00245-022-09962-0
Popis:	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. Comment: 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
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2206.02774 Zobrazit plný text záznamu Plný text ve formátu PDF