Sampling from the Mean-Field Stationary Distribution
<|reference_start|>Sampling from the Mean-Field Stationary Distribution: 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. A key technical contribution is to establish a new uniform-in-$N$ log-Sobolev inequality for the stationary distribution of the mean-field Langevin dynamics.<|reference_end|>
@article{kook2024sampling,
    title={Sampling from the Mean-Field Stationary Distribution},
    author={Yunbum Kook, Matthew S. Zhang, Sinho Chewi, Murat A. Erdogdu, Mufan
  Bill Li},
    journal={arXiv preprint arXiv:2402.07355},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.07355},
    primaryClass={math.ST cs.LG stat.ML stat.TH}
}