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

The Polyak-Lojasiewicz inequality (PLI) in ℝd 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 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 PLI for a broad class of such problems for energy functions regularised by the KL-divergence.