Position: Many generalization measures for deep learning are fragile

Abstract

A wide variety of generalization measures have been applied to deep neural networks (DNNs). Although obtaining tight bounds remains challenging, such measures are often assumed to reproduce qualitative generalization trends. In this position paper, we argue that many post-mortem generalization measures -- those computed on trained networks -- are \textbf{fragile}: small training modifications that barely affect the underlying DNN can substantially change a measure's value, trend, or scaling behavior. For example, minor hyperparameter changes, such as learning rate adjustments or switching between SGD variants can reverse the slope of a learning curve in widely used generalization measures like the path norm. We also identify subtler forms of fragility. For instance, the PAC-Bayes origin measure is regarded as one of the most reliable, and is indeed less sensitive to hyperparameter tweaks than many other measures. However, it completely fails to capture differences in data complexity across learning curves. This data fragility contrasts with the function-based marginal-likelihood PAC-Bayes bound, which does capture differences in data-complexity, including scaling behavior, in learning curves, but which is not a post-mortem measure. Beyond demonstrating that many bounds -- such as path, spectral and Frobenius norms, flatness proxies, and deterministic PAC-Bayes surrogates -- are fragile, this position paper also argues that developers of new measures should explicitly audit them for fragility.