KL-UCB-switch: optimal regret bounds for stochastic bandits from both a distribution-dependent and a distribution-free viewpoints - Archive ouverte HAL Access content directly
Journal Articles Journal of Machine Learning Research Year : 2022

## KL-UCB-switch: optimal regret bounds for stochastic bandits from both a distribution-dependent and a distribution-free viewpoints

Aurélien Garivier
Pierre Ménard
• Function : Author
• PersonId : 1022182
Gilles Stoltz

#### Abstract

We consider $K$-–armed stochastic bandits and consider cumulative regret bounds up to time $T$. We are interested in strategies achieving \emph{simultaneously} a distribution-free regret bound of optimal order $\sqrt{KT}$ and a distribution-dependent regret that is asymptotically optimal, that is, matching the $\kappa\ln T$ lower bound by Lai and Robbins (1985) and Burnetas and Katehakis (1996), where $\kappa$ is the optimal problem-dependent constant. This constant $\kappa$ depends on the model $\mathcal{D}$ considered (the family of possible distributions over the arms). M{\'e}nard and Garivier (2017) provided strategies achieving such a bi-optimality in the parametric case of models given by one-dimensional exponential families, while Lattimore (2016, 2018) did so for the family of (sub)Gaussian distributions with variance less than $1$. We extend this result to the non-parametric case of all distributions over $[0,1]$. We do so by combining the MOSS strategy by Audibert and Bubeck (2009), which enjoys a distribution-free regret bound of optimal order $\sqrt{KT}$, and the KL-UCB strategy by Capp{\'e} et al. (2013), for which we provide in passing the first analysis of an optimal distributiondependent $\kappa\ln T$ regret bound in the model of all distributions over $[0,1]$. We were able to obtain this non-parametric bi-optimality result while working hard to streamline the proofs (of previously known regret bounds and thus of the new analyses carried out); a second merit of the present contribution is therefore to provide a review of proofs of classical regret bounds for index-based strategies for K–armed stochastic bandits.

### Dates and versions

hal-01785705 , version 1 (04-05-2018)
hal-01785705 , version 2 (05-11-2019)
hal-01785705 , version 3 (28-06-2022)

### Identifiers

• HAL Id : hal-01785705 , version 3

### Cite

Aurélien Garivier, Hédi Hadiji, Pierre Ménard, Gilles Stoltz. KL-UCB-switch: optimal regret bounds for stochastic bandits from both a distribution-dependent and a distribution-free viewpoints. Journal of Machine Learning Research, In press. ⟨hal-01785705v3⟩

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