Ergodicity of the underdamped mean-field Langevin dynamics
This work addresses convergence issues in optimization algorithms for machine learning, particularly in overparametrized settings, but appears incremental as it extends existing mean-field Langevin dynamics analysis.
The paper analyzes the long-term behavior of the underdamped mean-field Langevin equation, establishing general and exponential convergence rates, and applies these results to study Hamiltonian gradient descent for overparametrized optimization, with a numerical example in training GANs.
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).