  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.
