Universal Approximation Constraints of Narrow ResNets: The Tunnel Effect

TL;DR

This paper investigates the limitations of narrow ResNets in universal function approximation, revealing a 'tunnel effect' where critical points shift to infinity, affecting their expressivity.

Contribution

It provides theoretical and numerical analysis of ResNet approximation constraints, introducing the 'tunnel effect' and deriving bounds based on channel ratios.

Findings

ResNets cannot represent critical points without input augmentation.

The 'tunnel effect' causes critical points to shift to infinity in certain regimes.

Approximation bounds depend explicitly on channel ratios and weight bounds.

Abstract

We analyze the universal approximation constraints of narrow Residual Neural Networks (ResNets) both theoretically and numerically. For deep neural networks without input space augmentation, a central constraint is the inability to represent critical points of the input-output map. We prove that this has global consequences for target function approximations and show that the manifestation of this defect is typically a shift of the critical point to infinity, which we call the ``tunnel effect'' in the context of classification tasks. While ResNets offer greater expressivity than standard multilayer perceptrons (MLPs), their capability strongly depends on the signal ratio between the skip and residual channels. We establish quantitative approximation bounds for both the residual-dominant (close to MLP) and skip-dominant (close to neural ODE) regimes. These estimates depend explicitly on the channel ratio and uniform network weight bounds. Low-dimensional examples further provide a detailed analysis of the different ResNet regimes and how architecture-target incompatibility influences the approximation error.