  We investigate the problem dependent regime in the stochastic Thresholding
Bandit problem (TBP) under several shape constraints. In the TBP, the objective
of the learner is to output, at the end of a sequential game, the set of arms
whose means are above a given threshold. The vanilla, unstructured, case is
already well studied in the literature. Taking $K$ as the number of arms, we
consider the case where (i) the sequence of arm's means $(\mu_k)_{k=1}^K$ is
monotonically increasing (MTBP) and (ii) the case where $(\mu_k)_{k=1}^K$ is
concave (CTBP). We consider both cases in the problem dependent regime and
study the probability of error - i.e. the probability to mis-classify at least
one arm. In the fixed budget setting, we provide upper and lower bounds for the
probability of error in both the concave and monotone settings, as well as
associated algorithms. In both settings the bounds match in the problem
dependent regime up to universal constants in the exponential.
