Projection-Free Online Convex Optimization with Time-Varying Constraints

Dan Garber, Ben Kretzu
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:14988-15005, 2024.

Abstract

We consider the setting of online convex optimization with adversarial time-varying constraints in which actions must be feasible w.r.t. a fixed constraint set, and are also required on average to approximately satisfy additional time-varying constraints. Motivated by scenarios in which the fixed feasible set (hard constraint) is difficult to project on, we consider projection-free algorithms that access this set only through a linear optimization oracle (LOO). We present an algorithm that, on a sequence of length $T$ and using overall $T$ calls to the LOO, guarantees $\tilde{O}(T^{3/4})$ regret w.r.t. the losses and $O(T^{7/8})$ constraints violation (ignoring all quantities except for $T$). In particular, these bounds hold w.r.t. any interval of the sequence. This algorithm however also requires access to an oracle for minimizing a strongly convex nonsmooth function over a Euclidean ball. We present a more efficient algorithm that does not require the latter optimization oracle but only first-order access to the time-varying constraints, and achieves similar bounds w.r.t. the entire sequence. We extend the latter to the setting of bandit feedback and obtain similar bounds (as a function of $T$) in expectation.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-garber24a, title = {Projection-Free Online Convex Optimization with Time-Varying Constraints}, author = {Garber, Dan and Kretzu, Ben}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {14988--15005}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/garber24a/garber24a.pdf}, url = {https://proceedings.mlr.press/v235/garber24a.html}, abstract = {We consider the setting of online convex optimization with adversarial time-varying constraints in which actions must be feasible w.r.t. a fixed constraint set, and are also required on average to approximately satisfy additional time-varying constraints. Motivated by scenarios in which the fixed feasible set (hard constraint) is difficult to project on, we consider projection-free algorithms that access this set only through a linear optimization oracle (LOO). We present an algorithm that, on a sequence of length $T$ and using overall $T$ calls to the LOO, guarantees $\tilde{O}(T^{3/4})$ regret w.r.t. the losses and $O(T^{7/8})$ constraints violation (ignoring all quantities except for $T$). In particular, these bounds hold w.r.t. any interval of the sequence. This algorithm however also requires access to an oracle for minimizing a strongly convex nonsmooth function over a Euclidean ball. We present a more efficient algorithm that does not require the latter optimization oracle but only first-order access to the time-varying constraints, and achieves similar bounds w.r.t. the entire sequence. We extend the latter to the setting of bandit feedback and obtain similar bounds (as a function of $T$) in expectation.} }
Endnote
%0 Conference Paper %T Projection-Free Online Convex Optimization with Time-Varying Constraints %A Dan Garber %A Ben Kretzu %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-garber24a %I PMLR %P 14988--15005 %U https://proceedings.mlr.press/v235/garber24a.html %V 235 %X We consider the setting of online convex optimization with adversarial time-varying constraints in which actions must be feasible w.r.t. a fixed constraint set, and are also required on average to approximately satisfy additional time-varying constraints. Motivated by scenarios in which the fixed feasible set (hard constraint) is difficult to project on, we consider projection-free algorithms that access this set only through a linear optimization oracle (LOO). We present an algorithm that, on a sequence of length $T$ and using overall $T$ calls to the LOO, guarantees $\tilde{O}(T^{3/4})$ regret w.r.t. the losses and $O(T^{7/8})$ constraints violation (ignoring all quantities except for $T$). In particular, these bounds hold w.r.t. any interval of the sequence. This algorithm however also requires access to an oracle for minimizing a strongly convex nonsmooth function over a Euclidean ball. We present a more efficient algorithm that does not require the latter optimization oracle but only first-order access to the time-varying constraints, and achieves similar bounds w.r.t. the entire sequence. We extend the latter to the setting of bandit feedback and obtain similar bounds (as a function of $T$) in expectation.
APA
Garber, D. & Kretzu, B.. (2024). Projection-Free Online Convex Optimization with Time-Varying Constraints. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:14988-15005 Available from https://proceedings.mlr.press/v235/garber24a.html.

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