  We consider $K$-armed stochastic bandits and consider cumulative regret
bounds up to time $T$. We are interested in strategies achieving 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\'enard 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 distribution-dependent
$\kappa\ln T$ 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.
