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

Linshan Liu; Mateusz B. Majka; Łukasz Szpruch

doi:10.1007/s00245-022-09962-0

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

Research output: Contribution to journal › Article › peer-review

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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 language	English
Article number	48
Journal	Applied Mathematics & Optimization
Volume	87
Issue number	3
Early online date	13 Mar 2023
DOIs	https://doi.org/10.1007/s00245-022-09962-0
Publication status	Published - Jun 2023

Keywords

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

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title = "Polyak–{\L}ojasiewicz inequality on the space of measures and convergence of mean-field birth-death processes",

abstract = "The Polyak–{\L}ojasiewicz inequality (P{\L}I) 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 P{\L}I 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 P{\L}I for a broad class of such problems for energy functions regularised by the KL-divergence.",

keywords = "Article, Mean-field optimization, Polyak–{\L}ojasiewicz condition, Exponential convergence, Birth-death processes, Fisher–Rao gradient flow, 49Q20",

author = "Linshan Liu and Majka, {Mateusz B.} and {\L}ukasz Szpruch",

note = "Funding Information: LS acknowledges the support of the UKRI Prosperity Partnership Scheme (FAIR) under EPSRC Grant EP/V056883/1. Publisher Copyright: {\textcopyright} 2023, The Author(s).",

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AU - Szpruch, Łukasz

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KW - Article

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KW - Exponential convergence

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KW - Fisher–Rao gradient flow

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

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