  Mean-field Langevin dynamics (MLFD) is a class of interacting particle
methods that tackle convex optimization over probability measures on a
manifold, which are scalable, versatile, and enjoy computational guarantees.
However, some important problems -- such as risk minimization for infinite
width two-layer neural networks, or sparse deconvolution -- are originally
defined over the set of signed, rather than probability, measures. In this
paper, we investigate how to extend the MFLD framework to convex optimization
problems over signed measures. Among two known reductions from signed to
probability measures -- the lifting and the bilevel approaches -- we show that
the bilevel reduction leads to stronger guarantees and faster rates (at the
price of a higher per-iteration complexity). In particular, we investigate the
convergence rate of MFLD applied to the bilevel reduction in the low-noise
regime and obtain two results. First, this dynamics is amenable to an annealing
schedule, adapted from Suzuki et al. (2023), that results in improved
convergence rates to a fixed multiplicative accuracy. Second, we investigate
the problem of learning a single neuron with the bilevel approach and obtain
local exponential convergence rates that depend polynomially on the dimension
and noise level (to compare with the exponential dependence that would result
from prior analyses).
