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

Linshan Liu, Mateusz B. Majka*, Łukasz Szpruch

*Corresponding author for this work

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Abstract

The Polyak–Łojasiewicz inequality (PŁI) in Rd is a natural condition for proving convergence of gradient descent algorithms (Karimi et al. in: Frasconi et al. (eds) Machine learning and knowledge discovery in databases, Springer International Publishing, Cham, pp 795–811, 2016). In the present paper, we study an analogue of PŁI 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 PŁI for a broad class of such problems for energy functions regularised by the KL-divergence.
Original languageEnglish
Article number48
JournalApplied Mathematics & Optimization
Volume87
Issue number3
Early online date13 Mar 2023
DOIs
Publication statusPublished - Jun 2023

Keywords

  • Article
  • Mean-field optimization
  • Polyak–Łojasiewicz condition
  • Exponential convergence
  • Birth-death processes
  • Fisher–Rao gradient flow
  • 49Q20

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