  Noisy particle gradient descent (NPGD) is an algorithm to minimize convex
functions over the space of measures that include an entropy term. In the
many-particle limit, this algorithm is described by a Mean-Field Langevin
dynamics - a generalization of the Langevin dynamics with a non-linear drift -
which is our main object of study. Previous work have shown its convergence to
the unique minimizer via non-quantitative arguments. We prove that this
dynamics converges at an exponential rate, under the assumption that a certain
family of Log-Sobolev inequalities holds. This assumption holds for instance
for the minimization of the risk of certain two-layer neural networks, where
NPGD is equivalent to standard noisy gradient descent. We also study the
annealed dynamics, and show that for a noise decaying at a logarithmic rate,
the dynamics converges in value to the global minimizer of the unregularized
objective function.
