Despite the widespread empirical success of ResNet, the generalization
properties of deep ResNet are rarely explored beyond the lazy training regime.
In this work, we investigate \emph{scaled} ResNet in the limit of infinitely
deep and wide neural networks, of which the gradient flow is described by a
partial differential equation in the large-neural network limit, i.e., the
\emph{mean-field} regime. To derive the generalization bounds under this
setting, our analysis necessitates a shift from the conventional time-invariant
Gram matrix employed in the lazy training regime to a time-variant,
distribution-dependent version. To this end, we provide a global lower bound on
the minimum eigenvalue of the Gram matrix under the mean-field regime. Besides,
for the traceability of the dynamic of Kullback-Leibler (KL) divergence, we
establish the linear convergence of the empirical error and estimate the upper
bound of the KL divergence over parameters distribution. Finally, we build the
uniform convergence for generalization bound via Rademacher complexity. Our
results offer new insights into the generalization ability of deep ResNet
beyond the lazy training regime and contribute to advancing the understanding
of the fundamental properties of deep neural networks.

在无穷深和宽神经网络的极限状态下，我们对被称为 scaled ResNet 的模型进行研究，以推导出其在 mean-field regime 下的泛化能力的界限，并提供了关于 Gram 矩阵最小特征值的全局下界、Kullback-Leibler 散度的线性收敛性以及 Rademacher 复杂度的统一收敛性，从而揭示了 lazy training regime 以外的深度神经网络通用化能力的新见解。