# Theoretical Convergence of Multi-Step Model-Agnostic Meta-Learning

arXiv: https://arxiv.org/abs/2002.07836 · HTML: https://mlsanity.com/paper/2002.07836/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Feb 2020.

## Built on by (most cited first, 28 of 28)

- [Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond](https://mlsanity.com/paper/2101.11517/) (2021)
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- [Generalization Bounds For Meta-Learning: An Information-Theoretic Analysis](https://mlsanity.com/paper/2109.14595/) (2021)
- [Generalization of Model-Agnostic Meta-Learning Algorithms: Recurring and Unseen Tasks](https://mlsanity.com/paper/2102.03832/) (2021)
- [Will Bilevel Optimizers Benefit from Loops](https://mlsanity.com/paper/2205.14224/) (2022)
- [On the Convergence Theory for Hessian-Free Bilevel Algorithms](https://mlsanity.com/paper/2110.07004/) (2021)
- [Tighter Analysis of Alternating Stochastic Gradient Method for Stochastic Nested Problems](https://mlsanity.com/paper/2106.13781/) (2021)
- [Global Convergence of MAML and Theory-Inspired Neural Architecture Search for Few-Shot Learning](https://mlsanity.com/paper/2203.09137/) (2022)
- [Optimal Algorithms for Stochastic Multi-Level Compositional Optimization](https://mlsanity.com/paper/2202.07530/) (2022)
- [Compositional federated learning: Applications in distributionally robust averaging and meta learning](https://mlsanity.com/paper/2106.11264/) (2021)
- [A Theoretical Understanding of Gradient Bias in Meta-Reinforcement Learning](https://mlsanity.com/paper/2112.15400/) (2021)
- [Meta-learning with negative learning rates](https://mlsanity.com/paper/2102.00940/) (2021)
- [Provable Generalization of Overparameterized Meta-learning Trained with SGD](https://mlsanity.com/paper/2206.09136/) (2022)
- [A Fast and Convergent Proximal Algorithm for Regularized Nonconvex and Nonsmooth Bi-level Optimization](https://mlsanity.com/paper/2203.16615/) (2022)
- [Memory-Based Optimization Methods for Model-Agnostic Meta-Learning and Personalized Federated Learning](https://mlsanity.com/paper/2106.04911/) (2021)
- [Unifying Gradient Estimators for Meta-Reinforcement Learning via Off-Policy Evaluation](https://mlsanity.com/paper/2106.13125/) (2021)
- [Biased Gradient Estimate with Drastic Variance Reduction for Meta Reinforcement Learning](https://mlsanity.com/paper/2112.07328/) (2021)
- [PersA-FL: Personalized Asynchronous Federated Learning](https://mlsanity.com/paper/2210.01176/) (2022)
- [Multi-level Monte-Carlo Gradient Methods for Stochastic Optimization with Biased Oracles](https://mlsanity.com/paper/2408.11084/) (2024)
- [Debiasing a First-order Heuristic for Approximate Bi-level Optimization](https://mlsanity.com/paper/2106.02487/) (2021)
- [Regularization for Adversarial Robust Learning](https://mlsanity.com/paper/2408.09672/) (2024)
- [On First-Order Meta-Reinforcement Learning with Moreau Envelopes](https://mlsanity.com/paper/2305.12216/) (2023)
- [Projection-Free Variance Reduction Methods for Stochastic Constrained Multi-Level Compositional Optimization](https://mlsanity.com/paper/2406.03787/) (2024)
- [Stability and Generalization for Stochastic Recursive Momentum-based Algorithms for (Strongly-)Convex One to K-Level Stochastic Optimizations](https://mlsanity.com/paper/2407.05286/) (2024)
- [Bounded Coupled AI Learning Dynamics in Tri-Hierarchical Drone Swarms](https://mlsanity.com/paper/2603.20333/) (2026)
- [Optimal Momentum Methods for Stochastic Multilevel Compositional Optimization](https://mlsanity.com/paper/2610.01572/) (2026)

## Abstract (arXiv)

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 ε-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.