Sampling from the Mean-Field Stationary Distribution

Yunbum Kook, Matthew S. Zhang, Sinho Chewi, Murat A. Erdogdu, Mufan (Bill) Li
Proceedings of Thirty Seventh Conference on Learning Theory, PMLR 247:3099-3136, 2024.

Abstract

We study the complexity of sampling from the stationary distribution of a mean-field SDE, or equivalently, the complexity of minimizing a functional over the space of probability measures which includes an interaction term. Our main insight is to decouple the two key aspects of this problem: (1) approximation of the mean-field SDE via a finite-particle system, via uniform-in-time propagation of chaos, and (2) sampling from the finite-particle stationary distribution, via standard log-concave samplers. Our approach is conceptually simpler and its flexibility allows for incorporating the state-of-the-art for both algorithms and theory. This leads to improved guarantees in numerous settings, including better guarantees for optimizing certain two-layer neural networks in the mean-field regime.

Cite this Paper

BibTeX
@InProceedings{pmlr-v247-kook24a, title = {Sampling from the Mean-Field Stationary Distribution}, author = {Kook, Yunbum and Zhang, Matthew S. and Chewi, Sinho and Erdogdu, Murat A. and Li, Mufan (Bill)}, booktitle = {Proceedings of Thirty Seventh Conference on Learning Theory}, pages = {3099--3136}, year = {2024}, editor = {Agrawal, Shipra and Roth, Aaron}, volume = {247}, series = {Proceedings of Machine Learning Research}, month = {30 Jun--03 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v247/kook24a/kook24a.pdf}, url = {https://proceedings.mlr.press/v247/kook24a.html}, abstract = { We study the complexity of sampling from the stationary distribution of a mean-field SDE, or equivalently, the complexity of minimizing a functional over the space of probability measures which includes an interaction term. Our main insight is to decouple the two key aspects of this problem: (1) approximation of the mean-field SDE via a finite-particle system, via uniform-in-time propagation of chaos, and (2) sampling from the finite-particle stationary distribution, via standard log-concave samplers. Our approach is conceptually simpler and its flexibility allows for incorporating the state-of-the-art for both algorithms and theory. This leads to improved guarantees in numerous settings, including better guarantees for optimizing certain two-layer neural networks in the mean-field regime.} }
Endnote
%0 Conference Paper %T Sampling from the Mean-Field Stationary Distribution %A Yunbum Kook %A Matthew S. Zhang %A Sinho Chewi %A Murat A. Erdogdu %A Mufan (Bill) Li %B Proceedings of Thirty Seventh Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2024 %E Shipra Agrawal %E Aaron Roth %F pmlr-v247-kook24a %I PMLR %P 3099--3136 %U https://proceedings.mlr.press/v247/kook24a.html %V 247 %X We study the complexity of sampling from the stationary distribution of a mean-field SDE, or equivalently, the complexity of minimizing a functional over the space of probability measures which includes an interaction term. Our main insight is to decouple the two key aspects of this problem: (1) approximation of the mean-field SDE via a finite-particle system, via uniform-in-time propagation of chaos, and (2) sampling from the finite-particle stationary distribution, via standard log-concave samplers. Our approach is conceptually simpler and its flexibility allows for incorporating the state-of-the-art for both algorithms and theory. This leads to improved guarantees in numerous settings, including better guarantees for optimizing certain two-layer neural networks in the mean-field regime.
APA
Kook, Y., Zhang, M.S., Chewi, S., Erdogdu, M.A. & Li, M.(.. (2024). Sampling from the Mean-Field Stationary Distribution. Proceedings of Thirty Seventh Conference on Learning Theory, in Proceedings of Machine Learning Research 247:3099-3136 Available from https://proceedings.mlr.press/v247/kook24a.html.

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