On the Convergence Theory of Gradient-Based Model-Agnostic Meta-Learning Algorithms.

In this paper, we study the convergence theory of a class of gradient-based Model-Agnostic Meta-Learning (MAML) methods and characterize their overall computational complexity as well as their best achievable level of accuracy in terms of gradient norm for nonconvex loss functions. In particular, we start with the MAML algorithm and its first order approximation (FO-MAML) and highlight the challenges that emerge in their analysis. By overcoming these challenges not only we provide the first theoretical guarantees for MAML and FO-MAML in nonconvex settings, but also we answer some of the unanswered questions for the implementation of these algorithms including how to choose their learning rate (stepsize) and the batch size for both tasks and datasets corresponding to tasks. In particular, we show that MAML can find an $\epsilon$-first-order stationary point for any $\epsilon$ after at most $\mathcal{O}(1/\epsilon^2)$ iterations while the cost of each iteration is $\mathcal{O}(d^2)$, where $d$ is the problem dimension. We further show that FO-MAML reduces the cost per iteration of MAML to $\mathcal{O}(d)$, but, unlike MAML, its solution cannot reach any small desired level of accuracy. We further propose a new variant of the MAML algorithm called Hessian-free MAML (HF-MAML) which preserves all theoretical guarantees of MAML, while reducing its computational cost per iteration from $\mathcal{O}(d^2)$ to $\mathcal{O}(d)$.