The role of invariance in spectral complexity-based generalization bounds

TL;DR

This paper investigates the limitations of spectral complexity measures in CNNs, revealing their insensitivity to invariances and their inability to differentiate between convolutional and locally connected architectures in terms of capacity bounds.

Contribution

It provides a theoretical and empirical analysis of spectral complexity, highlighting its insensitivity to CNN invariances and its limitations in accurately bounding network capacity.

Findings

Spectral complexity measures are insensitive to CNN invariances.

Spectral complexity yields similar bounds for convolutional and locally connected networks.

Limitations of spectral complexity challenge its use as a capacity measure.

Abstract

Deep convolutional neural networks (CNNs) have been shown to be able to fit a random labeling over data while still being able to generalize well for normal labels. Describing CNN capacity through a posteriori measures of complexity has been recently proposed to tackle this apparent paradox. These complexity measures are usually validated by showing that they correlate empirically with GE; being empirically larger for networks trained on random vs normal labels. Focusing on the case of spectral complexity we investigate theoretically and empirically the insensitivity of the complexity measure to invariances relevant to CNNs, and show several limitations of spectral complexity that occur as a result. For a specific formulation of spectral complexity we show that it results in the same upper bound complexity estimates for convolutional and locally connected architectures (which don't have the same favorable invariance properties). This is contrary to common intuition and empirical results.