Adaptation to the Range in $K$-Armed Bandits

Hédi Hadiji (1, 2) , Gilles Stoltz (1, 2)

1 LMO - Laboratoire de Mathématiques d'Orsay
2 CELESTE - Statistique mathématique et apprentissage

We consider stochastic bandit problems with $K$ arms, each associated with a bounded distribution supported on the range $[m,M]$. We do not assume that the range $[m,M]$ is known and show that there is a cost for learning this range. Indeed, a new trade-off between distribution-dependent and distribution-free regret bounds arises, which prevents from simultaneously achieving the typical $\ln T$ and $\sqrt{T}$ bounds. For instance, a $\sqrt{T}$}distribution-free regret bound may only be achieved if the distribution-dependent regret bounds are at least of order $\sqrt{T}$. We exhibit a strategy achieving the rates for regret indicated by the new trade-off.

Keywords: Multiarmed bandits; Adversarial learning; Cumulative regret; Information-theoretic proof techniques; multiarmed bandits; adversarial learning; cumulative regret; information-theoretic proof techniques

Domains: Machine Learning [stat.ML]; Statistics [math.ST]

Hédi Hadiji, Gilles Stoltz. Adaptation to the Range in $K$-Armed Bandits. Journal of Machine Learning Research, 2023, 24 (13), pp.1-33. ⟨hal-02794382v3⟩