  Strong student models can learn from weaker teachers: when trained on the
predictions of a weaker model, a strong pretrained student can learn to correct
the weak model's errors and generalize to examples where the teacher is not
confident, even when these examples are excluded from training. This enables
learning from cheap, incomplete, and possibly incorrect label information, such
as coarse logical rules or the generations of a language model. We show that
existing weak supervision theory fails to account for both of these effects,
which we call pseudolabel correction and coverage expansion, respectively. We
give a new bound based on expansion properties of the data distribution and
student hypothesis class that directly accounts for pseudolabel correction and
coverage expansion. Our bounds capture the intuition that weak-to-strong
generalization occurs when the strong model is unable to fit the mistakes of
the weak teacher without incurring additional error. We show that these
expansion properties can be checked from finite data and give empirical
evidence that they hold in practice.
