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.
Categories: cs.LG