Sampling from the Mean-Field Stationary Distribution

• URL: http://arxiv.org/abs/2402.07355v4
• Date: Fri, 5 Jul 2024 15:15:36 GMT
• Title: Sampling from the Mean-Field Stationary Distribution
• Abstract summary: We study the complexity of sampling from the stationary distribution of a mean-field SDE. Our approach is conceptually simpler and its flexibility allows for incorporating the state-of-the-art for both algorithms and theory. 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.
• Score: 20.953170955291522
• License: http://creativecommons.org/licenses/by/4.0/
• 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. 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.

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