  We are motivated by the problem of providing strong generalization guarantees
in the context of meta-learning. Existing generalization bounds are either
challenging to evaluate or provide vacuous guarantees in even relatively simple
settings. We derive a probably approximately correct (PAC) bound for
gradient-based meta-learning using two different generalization frameworks in
order to deal with the qualitatively different challenges of generalization at
the "base" and "meta" levels. We employ bounds for uniformly stable algorithms
at the base level and bounds from the PAC-Bayes framework at the meta level.
The result is a PAC-bound that is tighter when the base learner adapts quickly,
which is precisely the goal of meta-learning. We show that our bound provides a
tighter guarantee than other bounds on a toy non-convex problem on the unit
sphere and a text-based classification example. We also present a practical
regularization scheme motivated by the bound in settings where the bound is
loose and demonstrate improved performance over baseline techniques.
