Fantastic Generalization Measures are Nowhere to be Found

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, none of these bounds are tight. For instance, 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 imply guarantees that can explain the remarkable performance of neural networks. This raises the question of whether tight generalization bounds are at all possible. We consider two types of generalization bounds common in the literature: (1) bounds that depend on the training set and the output of the learning algorithm. There are multiple bounds of this type in the literature (e.g., norm-based and margin-based bounds), but we prove mathematically that no such bound can be uniformly tight in the overparameterized setting; (2) bounds that depend on the training set and 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 in the overparameterized setting, then no generalization bound can be tight for it. We conclude that generalization bounds in the overparameterized setting cannot be tight without suitable assumptions on the population distribution.