Convergence of mean-field Langevin dynamics: Time and space
  discretization, stochastic gradient, and variance reduction
Taiji Suzuki Denny Wu Atsushi Nitanda
The mean-field Langevin dynamics (MFLD) is a nonlinear generalization of the
Langevin dynamics that incorporates a distribution-dependent drift, and it
naturally arises from the optimization of two-layer neural networks via (noisy)
gradient descent. Recent works have shown that MFLD globally minimizes an
entropy-regularized convex functional in the space of measures. However, all
prior analyses assumed the infinite-particle or continuous-time limit, and
cannot handle stochastic gradient updates. We provide an general framework to
prove a uniform-in-time propagation of chaos for MFLD that takes into account
the errors due to finite-particle approximation, time-discretization, and
stochastic gradient approximation. To demonstrate the wide applicability of
this framework, we establish quantitative convergence rate guarantees to the
regularized global optimal solution under (i) a wide range of learning problems
such as neural network in the mean-field regime and MMD minimization, and (ii)
different gradient estimators including SGD and SVRG. Despite the generality of
our results, we achieve an improved convergence rate in both the SGD and SVRG
settings when specialized to the standard Langevin dynamics.