Matroid Semi-Bandits in Sublinear Time
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914814703435776 |
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| author | Tzeng, Ruo-Chun Ohsaka, Naoto Ariu, Kaito |
| author_facet | Tzeng, Ruo-Chun Ohsaka, Naoto Ariu, Kaito |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_17968 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Matroid Semi-Bandits in Sublinear Time Tzeng, Ruo-Chun Ohsaka, Naoto Ariu, Kaito Machine Learning 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. |
| title | Matroid Semi-Bandits in Sublinear Time |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2405.17968 |