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

arXiv: https://arxiv.org/abs/2301.06806 · HTML: https://mlsanity.com/paper/2301.06806/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 50 references that are in the map.
- First posted on arXiv 17 Jan 2023.

## Builds on (most cited first, 22 of 22)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [On First-Order Meta-Learning Algorithms](https://mlsanity.com/paper/1803.02999/) (2018)
- [Meta-Learning in Neural Networks: A Survey](https://mlsanity.com/paper/2004.05439/) (2020)
- [Personalized Federated Learning with Moreau Envelopes](https://mlsanity.com/paper/2006.08848/) (2020)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Sparsified SGD with Memory](https://mlsanity.com/paper/1809.07599/) (2018)
- [How to train your MAML](https://mlsanity.com/paper/1810.09502/) (2018)
- [Meta-Learning: A Survey](https://mlsanity.com/paper/1810.03548/) (2018)
- [SGD: General Analysis and Improved Rates](https://mlsanity.com/paper/1901.09401/) (2019)
- [Bayesian Model-Agnostic Meta-Learning](https://mlsanity.com/paper/1806.03836/) (2018)
- [Adaptive Gradient-Based Meta-Learning Methods](https://mlsanity.com/paper/1906.02717/) (2019)
- [On the Convergence Theory of Gradient-Based Model-Agnostic Meta-Learning Algorithms](https://mlsanity.com/paper/1908.10400/) (2019)
- [Lower Bounds and Optimal Algorithms for Personalized Federated Learning](https://mlsanity.com/paper/2010.02372/) (2020)
- [Random Reshuffling: Simple Analysis with Vast Improvements](https://mlsanity.com/paper/2006.05988/) (2020)
- [Perturbed Iterate Analysis for Asynchronous Stochastic Optimization](https://mlsanity.com/paper/1507.06970/) (2015)
- [Provable Guarantees for Gradient-Based Meta-Learning](https://mlsanity.com/paper/1902.10644/) (2019)
- [Torchmeta: A Meta-Learning library for PyTorch](https://mlsanity.com/paper/1909.06576/) (2019)
- [How Important is the Train-Validation Split in Meta-Learning?](https://mlsanity.com/paper/2010.05843/) (2020)
- [Complexity of Finding Stationary Points of Nonsmooth Nonconvex Functions](https://mlsanity.com/paper/2002.04130/) (2020)
- [A Distribution-Dependent Analysis of Meta-Learning](https://mlsanity.com/paper/2011.00344/) (2020)
- [Strongly convex functions, Moreau envelopes and the generic nature of convex functions with strong minimizers](https://mlsanity.com/paper/1507.07144/) (2015)

## Built on by (most cited first, 8 of 8)

- [Near-Optimal Nonconvex-Strongly-Convex Bilevel Optimization with Fully First-Order Oracles](https://mlsanity.com/paper/2306.14853/) (2023)
- [The Ball-Proximal (="Broximal") Point Method: a New Algorithm, Convergence Theory, and Applications](https://mlsanity.com/paper/2502.02002/) (2025)
- [On the Convergence of FedProx with Extrapolation and Inexact Prox](https://mlsanity.com/paper/2410.01410/) (2024)
- [Stochastic Constrained Decentralized Optimization for Machine Learning with Fewer Data Oracles: a Gradient Sliding Approach](https://mlsanity.com/paper/2404.02511/) (2024)
- [Understanding the Statistical Accuracy-Communication Trade-off in Personalized Federated Learning with Minimax Guarantees](https://mlsanity.com/paper/2410.08934/) (2024)
- [Natural Hypergradient Descent: Algorithm Design, Convergence Analysis, and Parallel Implementation](https://mlsanity.com/paper/2602.10905/) (2026)
- [Stability beyond Bounded Differences: Sharp Generalization Bounds under Finite Lp Moments](https://mlsanity.com/paper/2606.06855/) (2026)
- [Personalized Federated Reinforcement Learning via Model-Agnostic Meta-Learning: Convergence of Exact and Hessian-Free Meta-Policy Gradients](https://mlsanity.com/paper/2609.22833/) (2026)

## Abstract (arXiv)

In this work, we consider the problem of minimizing the sum of Moreau envelopes of given functions, which has previously appeared in the context of meta-learning and personalized federated learning. In contrast to the existing theory that requires running subsolvers until a certain precision is reached, we only assume that a finite number of gradient steps is taken at each iteration. As a special case, our theory allows us to show the convergence of First-Order Model-Agnostic Meta-Learning (FO-MAML) to the vicinity of a solution of Moreau objective. We also study a more general family of first-order algorithms that can be viewed as a generalization of FO-MAML. Our main theoretical achievement is a theoretical improvement upon the inexact SGD framework. In particular, our perturbed-iterate analysis allows for tighter guarantees that improve the dependency on the problem's conditioning. In contrast to the related work on meta-learning, ours does not require any assumptions on the Hessian smoothness, and can leverage smoothness and convexity of the reformulation based on Moreau envelopes. Furthermore, to fill the gaps in the comparison of FO-MAML to the Implicit MAML (iMAML), we show that the objective of iMAML is neither smooth nor convex, implying that it has no convergence guarantees based on the existing theory.