Fantastic Generalization Measures are Nowhere to be Found

TL;DR

This paper demonstrates that uniformly tight generalization bounds are impossible in overparameterized neural networks, providing formal proofs and highlighting a trade-off between algorithm performance and bound tightness.

Contribution

It proves mathematically that no uniformly tight generalization bounds exist for overparameterized models, and shows a trade-off between algorithm accuracy and bound tightness.

Findings

No uniformly tight bounds exist for overparameterized models.

A trade-off exists between model accuracy and the tightness of generalization bounds.

Formal proofs establish limitations of current generalization bounds.

Abstract

We study the notion of a generalization bound being uniformly tight, meaning that the difference between the bound and the population loss is small for all learning algorithms and all population distributions. Numerous generalization bounds have been proposed in the literature as potential explanations for the ability of neural networks to generalize in the overparameterized setting. However, in their paper ``Fantastic Generalization Measures and Where to Find Them,'' Jiang et al. (2020) examine more than a dozen generalization bounds, and show empirically that none of them are uniformly tight. This raises the question of whether uniformly-tight generalization bounds are at all possible in the overparameterized setting. We consider two types of generalization bounds: (1) bounds that may depend on the training set and the learned hypothesis (e.g., margin bounds). We prove mathematically that no such bound can be uniformly tight in the overparameterized setting; (2) bounds that may in addition also depend on the learning algorithm (e.g., stability bounds). For these bounds, we show a trade-off between the algorithm's performance and the bound's tightness. Namely, if the algorithm achieves good accuracy on certain distributions, then no generalization bound can be uniformly tight for it in the overparameterized setting. We explain how these formal results can, in our view, inform research on generalization bounds for neural networks, while stressing that other interpretations of these results are also possible.