Abstract
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 some numerical examples of the algorithm to train a generative adversarial network (GAN).
Funding Statement
The second author was supported in part by Finance For Energy Market Research Centre.
The third author was supported by Hong Kong RGC General Research Fund (project 14302921).
Acknowledgments
The authors wish to express their sincere appreciation to all those who made suggestions for improvements to this paper, in particular the four anonymous referees whose comments and suggestions helped us tremendously.
Citation
Anna Kazeykina. Zhenjie Ren. Xiaolu Tan. Junjian Yang. "Ergodicity of the underdamped mean-field Langevin dynamics." Ann. Appl. Probab. 34 (3) 3181 - 3226, June 2024. https://doi.org/10.1214/23-AAP2036
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