  Aimed at explaining the surprisingly good generalization behavior of
overparameterized deep networks, recent works have developed a variety of
generalization bounds for deep learning, all based on the fundamental
learning-theoretic technique of uniform convergence. While it is well-known
that many of these existing bounds are numerically large, through numerous
experiments, we bring to light a more concerning aspect of these bounds: in
practice, these bounds can {\em increase} with the training dataset size.
Guided by our observations, we then present examples of overparameterized
linear classifiers and neural networks trained by gradient descent (GD) where
uniform convergence provably cannot "explain generalization" -- even if we take
into account the implicit bias of GD {\em to the fullest extent possible}. More
precisely, even if we consider only the set of classifiers output by GD, which
have test errors less than some small $\epsilon$ in our settings, we show that
applying (two-sided) uniform convergence on this set of classifiers will yield
only a vacuous generalization guarantee larger than $1-\epsilon$. Through these
findings, we cast doubt on the power of uniform convergence-based
generalization bounds to provide a complete picture of why overparameterized
deep networks generalize well.
