Convex Analysis of the Mean Field Langevin Dynamics

As an example of the nonlinear Fokker-Planck equation, the \textit{mean field Langevin dynamics} recently attracts attention due to its connection to (noisy) gradient descent on infinitely wide neural networks in the mean field regime, and hence the convergence property of the dynamics is of great theoretical interest. In this work, we give a concise and self-contained convergence rate analysis of the mean field Langevin dynamics with respect to the (regularized) objective function in both continuous and discrete time settings. The key ingredient of our proof is a \textit{proximal Gibbs distribution} $p_q$ associated with the dynamics, which, in combination with techniques in \cite{vempala2019rapid}, allows us to develop a simple convergence theory parallel to classical results in convex optimization. Furthermore, we reveal that $p_q$ connects to the \textit{duality gap} in the empirical risk minimization setting, which enables efficient empirical evaluation of the algorithm convergence.