从平均场稳态分布中采样
Sampling from the Mean-Field Stationary Distribution
AI总结：
本文通过解耦有限粒子近似与对数凹采样，提出从平均场SDE稳态分布采样的新方法，并建立关于粒子数一致的对数Sobolev不等式，改进了平均场情形下两层神经网络优化等任务的保证。
摘要：
我们研究从平均场随机微分方程（SDE）的稳态分布中采样的复杂度，等价地，研究在包含交互项的概率测度空间上最小化一个泛函的复杂度。我们的核心见解是将该问题的两个关键方面解耦：（1）通过有限粒子系统近似平均场 SDE，借助时间一致的混沌传播；（2）通过标准对数凹采样器从有限粒子稳态分布中采样。我们的方法在概念上更简单，其灵活性允许同时融入算法和理论方面的最新成果。这在众多场景下带来了改进的保证，包括在平均场机制下优化某些两层神经网络的更优保证。一项关键的技术贡献是为平均场 Langevin 动力学的稳态分布建立了一个新的关于 $N$ 一致的对数 Sobolev 不等式。
英文摘要：
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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