  We study the complexity of sampling from the stationary distribution of a
mean-field SDE, or equivalently, the complexity of minimizing a functional over
the space of probability measures which includes an interaction term. Our main
insight is to decouple the two key aspects of this problem: (1) approximation
of the mean-field SDE via a finite-particle system, via uniform-in-time
propagation of chaos, and (2) sampling from the finite-particle stationary
distribution, via standard log-concave samplers. Our approach is conceptually
simpler and its flexibility allows for incorporating the state-of-the-art for
both algorithms and theory. This leads to improved guarantees in numerous
settings, including better guarantees for optimizing certain two-layer neural
networks in the mean-field regime. A key technical contribution is to establish
a new uniform-in-$N$ log-Sobolev inequality for the stationary distribution of
the mean-field Langevin dynamics.
