On the Generalization Behavior of Deep Residual Networks From a Dynamical System Perspective

TL;DR

This paper establishes generalization error bounds for deep residual networks using a dynamical system perspective, providing insights into their learning behavior and theoretical guarantees.

Contribution

It introduces unified generalization bounds for both discrete- and continuous-time ResNets based on dynamical systems theory and Rademacher complexity.

Findings

Generalization bounds of order O(1/√S) with respect to training samples

Depth-uniform and asymptotic generalization bounds under mild assumptions

Unified understanding of ResNets' generalization across different formulations

Abstract

Deep neural networks (DNNs) have significantly advanced machine learning, with model depth playing a central role in their successes. The dynamical system modeling approach has recently emerged as a powerful framework, offering new mathematical insights into the structure and learning behavior of DNNs. In this work, we establish generalization error bounds for both discrete- and continuous-time residual networks (ResNets) by combining Rademacher complexity, flow maps of dynamical systems, and the convergence behavior of ResNets in the deep-layer limit. The resulting bounds are of order $O(1/\sqrt{S})$ with respect to the number of training samples $S$, and include a structure-dependent negative term, yielding depth-uniform and asymptotic generalization bounds under milder assumptions. These findings provide a unified understanding of generalization across both discrete- and continuous-time ResNets, helping to close the gap in both the order of sample complexity and assumptions between the discrete- and continuous-time settings.