Projection-free Adaptive Regret with Membership Oracles

Zhou Lu, Nataly Brukhim, Paula Gradu, Elad Hazan
Proceedings of The 34th International Conference on Algorithmic Learning Theory, PMLR 201:1055-1073, 2023.

Abstract

In the framework of online convex optimization, most iterative algorithms require the computation of projections onto convex sets, which can be computationally expensive. To tackle this problem HK12 proposed the study of projection-free methods that replace projections with less expensive computations. The most common approach is based on the Frank-Wolfe method, that uses linear optimization computation in lieu of projections. Recent work by GK22 gave sublinear adaptive regret guarantees with projection free algorithms based on the Frank Wolfe approach. In this work we give projection-free algorithms that are based on a different technique, inspired by Mhammedi22, that replaces projections by set-membership computations. We propose a simple lazy gradient-based algorithm with a Minkowski regularization that attains near-optimal adaptive regret bounds. For general convex loss functions we improve previous adaptive regret bounds from $O(T^{3/4})$ to $O(\sqrt{T})$, and further to tight interval dependent bound $\tilde{O}(\sqrt{I})$ where $I$ denotes the interval length. For strongly convex functions we obtain the first poly-logarithmic adaptive regret bounds using a projection-free algorithm.

Cite this Paper

BibTeX
@InProceedings{pmlr-v201-lu23a, title = {Projection-free Adaptive Regret with Membership Oracles}, author = {Lu, Zhou and Brukhim, Nataly and Gradu, Paula and Hazan, Elad}, booktitle = {Proceedings of The 34th International Conference on Algorithmic Learning Theory}, pages = {1055--1073}, year = {2023}, editor = {Agrawal, Shipra and Orabona, Francesco}, volume = {201}, series = {Proceedings of Machine Learning Research}, month = {20 Feb--23 Feb}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v201/lu23a/lu23a.pdf}, url = {https://proceedings.mlr.press/v201/lu23a.html}, abstract = {In the framework of online convex optimization, most iterative algorithms require the computation of projections onto convex sets, which can be computationally expensive. To tackle this problem HK12 proposed the study of projection-free methods that replace projections with less expensive computations. The most common approach is based on the Frank-Wolfe method, that uses linear optimization computation in lieu of projections. Recent work by GK22 gave sublinear adaptive regret guarantees with projection free algorithms based on the Frank Wolfe approach. In this work we give projection-free algorithms that are based on a different technique, inspired by Mhammedi22, that replaces projections by set-membership computations. We propose a simple lazy gradient-based algorithm with a Minkowski regularization that attains near-optimal adaptive regret bounds. For general convex loss functions we improve previous adaptive regret bounds from $O(T^{3/4})$ to $O(\sqrt{T})$, and further to tight interval dependent bound $\tilde{O}(\sqrt{I})$ where $I$ denotes the interval length. For strongly convex functions we obtain the first poly-logarithmic adaptive regret bounds using a projection-free algorithm. } }
Endnote
%0 Conference Paper %T Projection-free Adaptive Regret with Membership Oracles %A Zhou Lu %A Nataly Brukhim %A Paula Gradu %A Elad Hazan %B Proceedings of The 34th International Conference on Algorithmic Learning Theory %C Proceedings of Machine Learning Research %D 2023 %E Shipra Agrawal %E Francesco Orabona %F pmlr-v201-lu23a %I PMLR %P 1055--1073 %U https://proceedings.mlr.press/v201/lu23a.html %V 201 %X In the framework of online convex optimization, most iterative algorithms require the computation of projections onto convex sets, which can be computationally expensive. To tackle this problem HK12 proposed the study of projection-free methods that replace projections with less expensive computations. The most common approach is based on the Frank-Wolfe method, that uses linear optimization computation in lieu of projections. Recent work by GK22 gave sublinear adaptive regret guarantees with projection free algorithms based on the Frank Wolfe approach. In this work we give projection-free algorithms that are based on a different technique, inspired by Mhammedi22, that replaces projections by set-membership computations. We propose a simple lazy gradient-based algorithm with a Minkowski regularization that attains near-optimal adaptive regret bounds. For general convex loss functions we improve previous adaptive regret bounds from $O(T^{3/4})$ to $O(\sqrt{T})$, and further to tight interval dependent bound $\tilde{O}(\sqrt{I})$ where $I$ denotes the interval length. For strongly convex functions we obtain the first poly-logarithmic adaptive regret bounds using a projection-free algorithm.
APA
Lu, Z., Brukhim, N., Gradu, P. & Hazan, E.. (2023). Projection-free Adaptive Regret with Membership Oracles. Proceedings of The 34th International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 201:1055-1073 Available from https://proceedings.mlr.press/v201/lu23a.html.

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