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Statistically Efficient, Polynomial-Time Algorithms for Combinatorial Semi-Bandits

Abstract : We consider combinatorial semi-bandits over a set X ⊂ {0, 1} d where rewards are uncorrelated across items. For this problem, the algorithm ESCB yields the smallest known regret bound R(T) = O d (ln m) 2 (ln T) ∆ min after T rounds, where m = max x ∈X 1 ⊤ x. However, ESCB has computational complexity O(|X|), which is typically exponential in d, and cannot be used in large dimensions. We propose the first algorithm that is both computationally and statistically efficient for this problem with regret R(T) = O d (ln m) 2 (ln T) ∆ min and computational asymptotic complexity O(δ −1 T poly(d)), where δ T is a function which vanishes arbitrarily slowly. Our approach involves carefully designing AESCB, an approximate version of ESCB with the same regret guarantees. We show that, whenever budgeted linear maximization over X can be solved up to a given approximation ratio, AESCB is implementable in polynomial time O(δ −1 T poly(d)) by repeatedly maximizing a linear function over X subject to a linear budget constraint, and showing how to solve these maximization problems efficiently. Additional algorithms, proofs and numerical experiments are given in the complete version of this work.
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https://hal.archives-ouvertes.fr/hal-03201526
Contributor : Thibaut Cuvelier Connect in order to contact the contributor
Submitted on : Monday, April 19, 2021 - 6:09:58 AM
Last modification on : Sunday, May 2, 2021 - 3:27:41 AM
Long-term archiving on: : Tuesday, July 20, 2021 - 6:10:39 PM

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Thibaut Cuvelier, Richard Combes, Eric Gourdin. Statistically Efficient, Polynomial-Time Algorithms for Combinatorial Semi-Bandits. SIGMETRICS 2021, ACM, Jun 2021, Virtual Event, China. ⟨10.1145/3410220.3453926⟩. ⟨hal-03201526⟩

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