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

By Alireza Fallah, Aryan Mokhtari, Asuman E. Ozdaglar. 2019

268 citations recorded by the source corpus.

We study the convergence of a class of gradient-based Model-Agnostic Meta-Learning (MAML) methods and characterize their overall complexity as well as their best achievable accuracy in terms of gradient norm for nonconvex loss functions. We start with the MAML method 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 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 ($\epsilon$-FOSP) for any positive $\epsilon$ after at most $\mathcal{O}(1/\epsilon^2)$ iterations at the expense of requiring second-order information. We also show that FO-MAML which ignores the second-order information required in the update of MAML cannot achieve any small desired level of accuracy, i.e., FO-MAML cannot find an $\epsilon$-FOSP for any $\epsilon>0$. We further propose a new variant of the MAML algorithm called Hessian-free MAML which preserves all theoretical guarantees of MAML, without requiring access to second-order information.

Subfields: Artificial Intelligence

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Citations in the corpus

On the Outsized Importance of Learning Rates in Local Update Methods (2020, methodology): “As such, our work is also closely related to that of Fallah et al. (2019), who demonstrated differences (and non-convergence issues) of various MAML algorithms.”
Personalized Federated Learning: A Meta-Learning Approach (2020, methodology): “Note that as in other MAML Algorithms (Finn et al., 2017; Fallah et al., 2019), the update in (8) which exploits the stochastic gradient estimation in (9) can be implemented in two levels: (i) First for each user i and each iteration t we perform the following update ˜ w ik +1 ,t = w ik +1 ,t − 1 −…”
Sharp-MAML: Sharpness-Aware Model-Agnostic Meta Learning (2022, background): “In addition, the existing analysis of MAML in (Fallah et al., 2020) requires the growing batch size and thus results in a suboptimal O( −3) sample complexity.”
Towards Personalized Federated Learning (2021, methodology): “To reduce computational overhead, the authors evaluated 2 forms of gradient approximations: (i) FOMAML [34], which replaces the gradient estimate with its first-order approximation where the Hessian term is ignored; and (ii) HF-MAML [38], which replaces the Hessian-vector product with gradient diffe”
Multi-Step Model-Agnostic Meta-Learning: Convergence and Improved Algorithms (2020, methodology): “1 Basic Assumptions We first make the following standard assumptions (Fallah et al., 2019; Rajeswaran et al., 2019).”
Personalized Federated Learning using Hypernetworks (2021, methodology): “While PFL methods like pFedME (Dinh et al., 2020) and Per-FedAvg (Fallah et al., 2020a) To evaluate pFedHN in this setting, we use the CIFAR10 dataset, with a total of 100 clients, of which 90 are used for training and 10 are held out novel clients.”
Meta-Learning with Neural Tangent Kernels (2021, unknown): “…(MAML) (Finn et al., 2017) is perhaps one of the most popular and ﬂexible ones, with a number of follow-up works such as (Nichol et al., 2018; Finn et al., 2018; Yao et al., 2019; Khodak et al., 2019a;b; Denevi et al., 2019; Fallah et al., 2020; Lee et al., 2020; Tripuraneni et al., 2020).”
When MAML Can Adapt Fast and How to Assist When It Cannot (2019, background): “Understanding how MAML and its alike work continues to draw research interests (Finn and Levine, 2018; Fallah et al., 2020; Raghu et al., 2020; Saunshi et al., 2020).”
Personalized Federated Learning with Theoretical Guarantees: A Model-Agnostic Meta-Learning Approach (2020, background): “Note that as in other MAML methods [2, 31], the update in (7) can be implemented in two stages: First, we compute w̃ k+1,t = w i k+1,t−1−α∇̃fi(w k+1,t−1,D t) and then evaluate w k+1,t by w k+1,t = w k+1 − β(I − α∇̃(2)fi(w k+1,t−1,D ′′i t ))∇̃fi(w̃ k+1,t,D ′i t ).”
Learned Initializations for Optimizing Coordinate-Based Neural Representations (2020, methodology): “Most relevant to this work are optimization-based metalearning algorithms such as Model-Agnostic Meta Learning (MAML) [6] and Reptile [25], as well as various extensions [1, 5, 7, 17, 29].”
A Sample Complexity Separation between Non-Convex and Convex Meta-Learning (2020, background): “While the optimization properties of gradient-based meta-learning algorithms have been recently studied in the non-convex setting [Fallah et al., 2019, Rajeswaran et al., 2019, Zhou et al., 2019], these results only provide stationary-point convergence guarantees and do not show a reduction in…”