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

Problem Dependent View on Structured Thresholding Bandit Problems