  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.
