  We study generalization properties of weakly supervised learning. That is,
learning where only a few "strong" labels (the actual target of our prediction)
are present but many more "weak" labels are available. In particular, we show
that having access to weak labels can significantly accelerate the learning
rate for the strong task to the fast rate of $\mathcal{O}(\nicefrac1n)$, where
$n$ denotes the number of strongly labeled data points. This acceleration can
happen even if by itself the strongly labeled data admits only the slower
$\mathcal{O}(\nicefrac{1}{\sqrt{n}})$ rate. The actual acceleration depends
continuously on the number of weak labels available, and on the relation
between the two tasks. Our theoretical results are reflected empirically across
a range of tasks and illustrate how weak labels speed up learning on the strong
task.
