The Influence of Shape Constraints on the Thresholding Bandit Problem

TL;DR

This paper studies how shape constraints like monotonicity, unimodality, and concavity affect the difficulty and optimal strategies for the Thresholding Bandit Problem, providing minimax regret rates for each case.

Contribution

It derives problem-independent minimax regret rates for TBP under various shape constraints and introduces algorithms tailored to each setting.

Findings

Minimax regret rates vary significantly with shape constraints.

Shape constraints fundamentally alter the TBP's complexity.

Provided algorithms achieve the derived minimax rates.

Abstract

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.