Abstract
We consider d-dimensional stochastic continuum-armed bandits with the expected reward function being additive β-Hölder with sparsity s for and . The rate of convergence for the minimax regret is established where T is the number of rounds. In particular, the minimax regret does not depend on d and is linear in s. A novel algorithm is proposed and is shown to be rate-optimal, up to a logarithmic factor of T.
The problem of adaptivity is also studied. A lower bound on the cost of adaptation to the smoothness is obtained and the result implies that adaptation for free is impossible in general without further structural assumptions. We then consider adaptive additive SCAB under an additional self-similarity assumption. An adaptive procedure is constructed and is shown to simultaneously achieve the minimax regret for a range of smoothness levels.
Funding Statement
The research was supported in part by NSF Grant DMS-2015259 and NIH Grants R01-GM129781 and R01-GM123056.
Acknowledgments
We would like to thank the Associate Editor and the referees for their detailed and constructive comments which have helped to improve the presentation of the paper.
Citation
T. Tony Cai. Hongming Pu. "Stochastic continuum-armed bandits with additive models: Minimax regrets and adaptive algorithm." Ann. Statist. 50 (4) 2179 - 2204, August 2022. https://doi.org/10.1214/22-AOS2182
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