We propose a novel framework for exploring weak and $L_2$ generalization errors of algorithms through the lens of differential calculus on the space of probability measures. Specifically, we consider the KL-regularized empirical risk minimization problem and establish generic conditions under which the generalization error convergence rate, when training on a sample of size $n$, is $\mathcal{O}(1/n)$. In the context of supervised learning with a one-hidden layer neural network in the mean-field regime, these conditions are reflected in suitable integrability and regularity assumptions on the loss and activation functions.

翻译：我们提出了一种新颖的框架，通过概率测度空间上的微分学视角，探讨算法的弱泛化误差与$L_2$泛化误差。具体而言，我们考虑KL正则化经验风险最小化问题，并建立了通用条件，使得在样本量为$n$的训练过程中，泛化误差的收敛速率为$\mathcal{O}(1/n)$。在平均场机制下，针对单隐层神经网络的监督学习场景，这些条件体现在对损失函数和激活函数的适当可积性及正则性假设中。