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The mean-field Langevin dynamics (MFLD) is a nonlinear generalization of theLangevin dynamics that incorporates a distribution-dependent drift, and itnaturally arises from the optimization of two-layer neural networks via (noisy)gradient descent. Recent works have shown that MFLD globally minimizes anentropy-regularized convex functional in the space of measures. However, allprior analyses assumed the infinite-particle or continuous-time limit, andcannot handle stochastic gradient updates. We provide an general framework toprove a uniform-in-time propagation of chaos for MFLD that takes into accountthe errors due to finite-particle approximation, time-discretization, andstochastic gradient approximation. To demonstrate the wide applicability ofthis framework, we establish quantitative convergence rate guarantees to theregularized global optimal solution under (i) a wide range of learning problemssuch as neural network in the mean-field regime and MMD minimization, and (ii)different gradient estimators including SGD and SVRG. Despite the generality ofour results, we achieve an improved convergence rate in both the SGD and SVRGsettings when specialized to the standard Langevin dynamics.

Convergence of mean-field Langevin dynamics: Time and space discretization, stochastic gradient, and variance reduction