The Influence of Shape Constraints on the Thresholding Bandit Problem

Cheshire, James; Ménard, Pierre; Carpentier, Alexandra

Document type :

Communication dans un congrès avec actes

Title :

The Influence of Shape Constraints on the Thresholding Bandit Problem

Author(s) :

Cheshire, James [Auteur]
Otto-von-Guericke-Universität Magdeburg = Otto-von-Guericke University [Magdeburg] [OVGU]
Ménard, Pierre [Auteur]
Scool [Scool]
Sequential Learning [SEQUEL]
Carpentier, Alexandra [Auteur]
Otto-von-Guericke-Universität Magdeburg = Otto-von-Guericke University [Magdeburg] [OVGU]

Conference title :

COLT 2020 - Thirty Third Conference on Learning Theory

City :

Graz / Virtual

Country :

Autriche

Start date of the conference :

2020-07-09

Publication date :

2020

HAL domain(s) :

Mathématiques [math]/Statistiques [math.ST]

English abstract : [en]

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 (µ k) k is monotonically ...
Show more >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 (µ k) k is monotonically increasing MTBP, (iii) the case where (µ k) k is unimodal UTBP and (iv) the case where (µ 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) log(K)K/T for TBP, (ii) log(K)/T for MTBP, (iii) K/T for UTBP and (iv) 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 problem to the other.Show less >

Language :

Anglais

Peer reviewed article :

Oui

Audience :

Internationale

Popular science :

Non

Collections :

Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL) - UMR 9189

Source :

Harvested from HAL

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