Matroid Semi-Bandits in Sublinear Time

Ruo-Chun Tzeng, Naoto Ohsaka, Kaito Ariu
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:48855-48877, 2024.

Abstract

We study the matroid semi-bandits problem, where at each round the learner plays a subset of $K$ arms from a feasible set, and the goal is to maximize the expected cumulative linear rewards. Existing algorithms have per-round time complexity at least $\Omega(K)$, which becomes expensive when $K$ is large. To address this computational issue, we propose FasterCUCB whose sampling rule takes time sublinear in $K$ for common classes of matroids: $\mathcal{O}(D\text{ polylog}(K)\text{ polylog}(T))$ for uniform matroids, partition matroids, and graphical matroids, and $\mathcal{O}(D\sqrt{K}\text{ polylog}(T))$ for transversal matroids. Here, $D$ is the maximum number of elements in any feasible subset of arms, and $T$ is the horizon. Our technique is based on dynamic maintenance of an approximate maximum-weight basis over inner-product weights. Although the introduction of an approximate maximum-weight basis presents a challenge in regret analysis, we can still guarantee an upper bound on regret as tight as CUCB in the sense that it matches the gap-dependent lower bound by Kveton et al. (2014a) asymptotically.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-tzeng24a, title = {Matroid Semi-Bandits in Sublinear Time}, author = {Tzeng, Ruo-Chun and Ohsaka, Naoto and Ariu, Kaito}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {48855--48877}, 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/tzeng24a/tzeng24a.pdf}, url = {https://proceedings.mlr.press/v235/tzeng24a.html}, abstract = {We study the matroid semi-bandits problem, where at each round the learner plays a subset of $K$ arms from a feasible set, and the goal is to maximize the expected cumulative linear rewards. Existing algorithms have per-round time complexity at least $\Omega(K)$, which becomes expensive when $K$ is large. To address this computational issue, we propose FasterCUCB whose sampling rule takes time sublinear in $K$ for common classes of matroids: $\mathcal{O}(D\text{ polylog}(K)\text{ polylog}(T))$ for uniform matroids, partition matroids, and graphical matroids, and $\mathcal{O}(D\sqrt{K}\text{ polylog}(T))$ for transversal matroids. Here, $D$ is the maximum number of elements in any feasible subset of arms, and $T$ is the horizon. Our technique is based on dynamic maintenance of an approximate maximum-weight basis over inner-product weights. Although the introduction of an approximate maximum-weight basis presents a challenge in regret analysis, we can still guarantee an upper bound on regret as tight as CUCB in the sense that it matches the gap-dependent lower bound by Kveton et al. (2014a) asymptotically.} }
Endnote
%0 Conference Paper %T Matroid Semi-Bandits in Sublinear Time %A Ruo-Chun Tzeng %A Naoto Ohsaka %A Kaito Ariu %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-tzeng24a %I PMLR %P 48855--48877 %U https://proceedings.mlr.press/v235/tzeng24a.html %V 235 %X We study the matroid semi-bandits problem, where at each round the learner plays a subset of $K$ arms from a feasible set, and the goal is to maximize the expected cumulative linear rewards. Existing algorithms have per-round time complexity at least $\Omega(K)$, which becomes expensive when $K$ is large. To address this computational issue, we propose FasterCUCB whose sampling rule takes time sublinear in $K$ for common classes of matroids: $\mathcal{O}(D\text{ polylog}(K)\text{ polylog}(T))$ for uniform matroids, partition matroids, and graphical matroids, and $\mathcal{O}(D\sqrt{K}\text{ polylog}(T))$ for transversal matroids. Here, $D$ is the maximum number of elements in any feasible subset of arms, and $T$ is the horizon. Our technique is based on dynamic maintenance of an approximate maximum-weight basis over inner-product weights. Although the introduction of an approximate maximum-weight basis presents a challenge in regret analysis, we can still guarantee an upper bound on regret as tight as CUCB in the sense that it matches the gap-dependent lower bound by Kveton et al. (2014a) asymptotically.
APA
Tzeng, R., Ohsaka, N. & Ariu, K.. (2024). Matroid Semi-Bandits in Sublinear Time. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:48855-48877 Available from https://proceedings.mlr.press/v235/tzeng24a.html.

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