  We study best-of-both-worlds algorithms for bandits with switching cost,
recently addressed by Rouyer, Seldin and Cesa-Bianchi, 2021. We introduce a
surprisingly simple and effective algorithm that simultaneously achieves
minimax optimal regret bound of $\mathcal{O}(T^{2/3})$ in the oblivious
adversarial setting and a bound of $\mathcal{O}(\min\{\log
(T)/\Delta^2,T^{2/3}\})$ in the stochastically-constrained regime, both with
(unit) switching costs, where $\Delta$ is the gap between the arms. In the
stochastically constrained case, our bound improves over previous results due
to Rouyer et al., that achieved regret of $\mathcal{O}(T^{1/3}/\Delta)$. We
accompany our results with a lower bound showing that, in general,
$\tilde{\Omega}(\min\{1/\Delta^2,T^{2/3}\})$ regret is unavoidable in the
stochastically-constrained case for algorithms with $\mathcal{O}(T^{2/3})$
worst-case regret.
