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

Alireza Fallah, Aryan Mokhtari, Asuman Ozdaglar
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:1082-1092, 2020.

Abstract

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 ?-first-order stationary point ( ?-FOSP) for any positive ? after at most O(1/?^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 ?-FOSP for any ?>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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v108-fallah20a, title = {On the Convergence Theory of Gradient-Based Model-Agnostic Meta-Learning Algorithms}, author = {Fallah, Alireza and Mokhtari, Aryan and Ozdaglar, Asuman}, booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics}, pages = {1082--1092}, year = {2020}, editor = {Chiappa, Silvia and Calandra, Roberto}, volume = {108}, series = {Proceedings of Machine Learning Research}, month = {26--28 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v108/fallah20a/fallah20a.pdf}, url = {https://proceedings.mlr.press/v108/fallah20a.html}, abstract = {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 ?-first-order stationary point ( ?-FOSP) for any positive ? after at most O(1/?^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 ?-FOSP for any ?>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.} }
Endnote
%0 Conference Paper %T On the Convergence Theory of Gradient-Based Model-Agnostic Meta-Learning Algorithms %A Alireza Fallah %A Aryan Mokhtari %A Asuman Ozdaglar %B Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2020 %E Silvia Chiappa %E Roberto Calandra %F pmlr-v108-fallah20a %I PMLR %P 1082--1092 %U https://proceedings.mlr.press/v108/fallah20a.html %V 108 %X 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 ?-first-order stationary point ( ?-FOSP) for any positive ? after at most O(1/?^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 ?-FOSP for any ?>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.
APA
Fallah, A., Mokhtari, A. & Ozdaglar, A.. (2020). On the Convergence Theory of Gradient-Based Model-Agnostic Meta-Learning Algorithms. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 108:1082-1092 Available from https://proceedings.mlr.press/v108/fallah20a.html.

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