Problem Dependent View on Structured Thresholding Bandit Problems

TL;DR

This paper explores the stochastic Thresholding Bandit problem under shape constraints, providing bounds and algorithms for monotonic and concave mean sequences in a fixed budget setting.

Contribution

It introduces problem-dependent bounds and algorithms for structured TBP with monotonic and concave mean sequences, extending the unstructured case.

Findings

Bounds match in the problem-dependent regime up to constants.

Provides algorithms with theoretical guarantees.

Analyzes probability of error in structured TBP settings.

Abstract

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.