Population-based learning paradigms, including evolutionary strategies, Population-Based Training (PBT), and recent model-merging methods, combine fast within-model optimisation with slower population-level adaptation. Despite their empirical success, a general mathematical description of the resulting collective training dynamics remains incomplete. We introduce a theoretical framework for neural network training based on two-time-scale population dynamics. We model a population of neural networks as an interacting agent system in which network parameters evolve through fast noisy gradient updates of SGD/Langevin type, while hyperparameters evolve through slower selection--mutation dynamics. We prove the large-population limit for the joint distribution of parameters and hyperparameters and, under strong time-scale separation, derive a selection--mutation equation for the hyperparameter density. For each fixed hyperparameter, the fast parameter dynamics relaxes to a Boltzmann--Gibbs measure, inducing an effective fitness for the slow evolution. The averaged dynamics connects population-based learning with bilevel optimisation and classical replicator--mutator models, yields conditions under which the population mean moves toward the fittest hyperparameter, and clarifies the role of noise and diversity in balancing optimisation and exploration. Numerical experiments illustrate both the large-population regime and the reduced two-time-scale dynamics, and indicate that access to the effective fitness, either in closed form or through population-level estimation, can improve population-level updates.

翻译：基于群体的学习范式，包括进化策略、群体训练（PBT）及近期提出的模型合并方法，将快速的模型内优化与较慢的群体层面自适应相结合。尽管这些方法在经验上取得了成功，但其产生的集体训练动态过程仍缺乏通用的数学描述。我们基于双时间尺度群体动力学，提出了一种神经网络训练的理论框架。我们将神经网络群体建模为一个相互作用的智能体系统，其中网络参数通过SGD/朗之万型的快速含噪梯度更新演化，而超参数则通过较慢的选择-突变动力学演化。我们证明了参数与超参数联合分布在群体规模趋于无穷时的极限，并在强时间尺度分离条件下推导出超参数密度的选择-突变方程。对于每个固定超参数，快速的参数动力学松弛为玻尔兹曼-吉布斯测度，从而为慢速演化引入了有效适应度。该平均化动力学将基于群体的学习与双层优化及经典的复制子-突变子模型联系起来，给出了群体均值向最优超参数移动的条件，并阐明了噪声与多样性在平衡优化与探索中的作用。数值实验验证了大规模群体行为与降阶的双时间尺度动力学，表明通过闭式解或群体级估计获取有效适应度，可改善群体层面的更新策略。