We study the long time behavior of an underdamped mean-field Langevin (MFL) equation, and provide a general convergence as well as an exponential convergence rate result under different conditions. The results on the MFL equation can be applied to study the convergence of the Hamiltonian gradient descent algorithm for the overparametrized optimization. We then provide a numerical example of the algorithm to train a generative adversarial networks (GAN).

翻译：我们研究了欠阻尼均场朗之万（MFL）方程的长时行为，并在不同条件下给出了一般收敛性以及指数收敛速率的结果。关于MFL方程的结果可应用于研究超参数化优化中哈密顿梯度下降算法的收敛性。随后，我们给出了该算法用于训练生成对抗网络（GAN）的数值示例。