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

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.