Generalization of Scaled Deep ResNets in the Mean-Field Regime
This work addresses the theoretical understanding of generalization in deep neural networks, specifically for ResNets, but is incremental as it extends existing analysis from lazy training to the mean-field regime.
The authors tackled the problem of understanding generalization in deep ResNets beyond lazy training by analyzing scaled ResNets in the mean-field regime, deriving generalization bounds through a time-variant Gram matrix and establishing convergence results for empirical error and KL divergence.
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.