We investigate the stochastic Thresholding Bandit problem (TBP) under several shape constraints. On top of (i) the vanilla, unstructured TBP, we consider the case where (ii) the sequence of arm's means $(\mu_k)_k$ is monotonically increasing MTBP, (iii) the case where $(\mu_k)_k$ is unimodal UTBP and (iv) the case where $(\mu_k)_k$ is concave CTBP. In the TBP problem the aim is to output, at the end of the sequential game, the set of arms whose means are above a given threshold. The regret is the highest gap between a misclassified arm and the threshold. In the fixed budget setting, we provide problem independent minimax rates for the expected regret in all settings, as well as associated algorithms. We prove that the minimax rates for the regret are (i) $\sqrt{\log(K)K/T}$ for TBP, (ii) $\sqrt{\log(K)/T}$ for MTBP, (iii) $\sqrt{K/T}$ for UTBP and (iv) $\sqrt{\log\log K/T}$ for CTBP, where $K$ is the number of arms and $T$ is the budget. These rates demonstrate that the dependence on $K$ of the minimax regret varies significantly depending on the shape constraint. This highlights the fact that the shape constraints modify fundamentally the nature of the TBP.

翻译：我们调查了几个形状限制下的盗匪问题(TBP)。在(一) 香草、无结构的TBP中,我们考虑的是:(二) 手臂的顺序意味着$(mu_k)_k美元,这是单质增加的MTBP,(三) 美元(mu_k)_k美元是单式UTBP,(四) 美元(mu_k)_k美元是同质的CTBP。在TBP问题中,目标是输出,在连续游戏结束时,一套手段超过给定阈值的武器。遗憾是错分类的手臂和阈值之间的最大差距。在固定的预算设置中,我们为所有环境中的预期遗憾提供独立的微缩税率,以及相关的算法。我们证明,对于TBP, 美元(k) 美元(K) K/TBP 的最小税率是(K) 美元(K) 和 美元(KBBP) 美元(K) 和 美元(K) 美元(K/T) 的硬度(K) 美元(K) 预算的缩数(K) 美元(K) 和美元/ 美元/美元) 美元(KBBPT) 的硬值是。