Generalization of Scaled Deep ResNets in the Mean-Field Regime
Yihang Chen Fanghui Liu Yiping Lu Grigorios G. Chrysos Volkan Cevher
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.