  As a popular meta-learning approach, the model-agnostic meta-learning (MAML)
algorithm has been widely used due to its simplicity and effectiveness.
However, the convergence of the general multi-step MAML still remains
unexplored. In this paper, we develop a new theoretical framework to provide
such convergence guarantee for two types of objective functions that are of
interest in practice: (a) resampling case (e.g., reinforcement learning), where
loss functions take the form in expectation and new data are sampled as the
algorithm runs; and (b) finite-sum case (e.g., supervised learning), where loss
functions take the finite-sum form with given samples. For both cases, we
characterize the convergence rate and the computational complexity to attain an
$\epsilon$-accurate solution for multi-step MAML in the general nonconvex
setting. In particular, our results suggest that an inner-stage stepsize needs
to be chosen inversely proportional to the number $N$ of inner-stage steps in
order for $N$-step MAML to have guaranteed convergence. From the technical
perspective, we develop novel techniques to deal with the nested structure of
the meta gradient for multi-step MAML, which can be of independent interest.
