Better Best of Both Worlds Bounds for Bandits with Switching Costs

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.