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

Abstract

The Polyak-Lojasiewicz inequality (P LI) in R d is a natural condition for proving convergence of gradient descent algorithms [13]. In the present paper, we study an analogue of P LI on the space of probability measures P(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 P LI for a broad class of such problems for energy functions regularised by the KL-divergence. 2020 Mathematics Subject Classification. 49Q20.