Mean-field Langevin dynamics (MFLD) is an optimization method derived by taking the mean-field limit of noisy gradient descent for two-layer neural networks in the mean-field regime. Recently, the propagation of chaos (PoC) for MFLD has gained attention as it provides a quantitative characterization of the optimization complexity in terms of the number of particles and iterations. A remarkable progress by Chen et al. (2022) showed that the approximation error due to finite particles remains uniform in time and diminishes as the number of particles increases. In this paper, by refining the defective log-Sobolev inequality -- a key result from that earlier work -- under the neural network training setting, we establish an improved PoC result for MFLD, which removes the exponential dependence on the regularization coefficient from the particle approximation term of the optimization complexity. As an application, we propose a PoC-based model ensemble strategy with theoretical guarantees.

翻译：平均场朗之万动力学（MFLD）是通过对平均场机制下双层神经网络的带噪梯度下降取平均场极限而导出的优化方法。近年来，MFLD的混沌传播（PoC）因其能以粒子数量和迭代次数定量刻画优化复杂度而受到关注。Chen等人（2022年）的显著进展表明，有限粒子数导致的近似误差在时间上保持均匀，并随粒子数增加而衰减。本文通过在神经网络训练场景下改进有缺陷的对数索伯列夫不等式——该早期工作的关键结果——为MFLD建立了改进的PoC结论，从而从优化复杂度的粒子近似项中消除了对正则化系数的指数依赖。作为应用，我们提出了一种具有理论保证的基于PoC的模型集成策略。