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In this work, we consider the problem of minimizing the sum of Moreauenvelopes of given functions, which has previously appeared in the context ofmeta-learning and personalized federated learning. In contrast to the existingtheory that requires running subsolvers until a certain precision is reached,we only assume that a finite number of gradient steps is taken at eachiteration. As a special case, our theory allows us to show the convergence ofFirst-Order Model-Agnostic Meta-Learning (FO-MAML) to the vicinity of asolution of Moreau objective. We also study a more general family offirst-order algorithms that can be viewed as a generalization of FO-MAML. Ourmain theoretical achievement is a theoretical improvement upon the inexact SGDframework. In particular, our perturbed-iterate analysis allows for tighterguarantees that improve the dependency on the problem's conditioning. Incontrast to the related work on meta-learning, ours does not require anyassumptions on the Hessian smoothness, and can leverage smoothness andconvexity of the reformulation based on Moreau envelopes. Furthermore, to fillthe gaps in the comparison of FO-MAML to the Implicit MAML (iMAML), we showthat the objective of iMAML is neither smooth nor convex, implying that it hasno convergence guarantees based on the existing theory.

Convergence of First-Order Algorithms for Meta-Learning with Moreau Envelopes