  Weight norm $\|w\|$ and margin $\gamma$ participate in learning theory via
the normalized margin $\gamma/\|w\|$. Since standard neural net optimizers do
not control normalized margin, it is hard to test whether this quantity
causally relates to generalization. This paper designs a series of experimental
studies that explicitly control normalized margin and thereby tackle two
central questions. First: does normalized margin always have a causal effect on
generalization? The paper finds that no -- networks can be produced where
normalized margin has seemingly no relationship with generalization, counter to
the theory of Bartlett et al. (2017). Second: does normalized margin ever have
a causal effect on generalization? The paper finds that yes -- in a standard
training setup, test performance closely tracks normalized margin. The paper
suggests a Gaussian process model as a promising explanation for this behavior.
