Matroid Semi-Bandits in Sublinear Time

Ruo-Chun Tzeng; Naoto Ohsaka; Kaito Ariu

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arXiv · 2405.17968

Matroid Semi-Bandits in Sublinear Time

Ruo-Chun Tzeng·Naoto Ohsaka·Kaito Ariu

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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 $Ω(K)$, 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: $O(D\text{ polylog}(K)\text{ polylog}(T))$ for uniform matroids, partition matroids, and graphical matroids, and $O(D\sqrt{K}\text{ polylog}(T))$ 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.

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BibTeX RIS

Ruo-Chun Tzeng, Naoto Ohsaka, Kaito Ariu. 2024-05-28. Matroid Semi-Bandits in Sublinear Time. https://arxiv.org/abs/2405.17968

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