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 [14]. We introduce a surprisingly simple and effective algorithm that simultaneously achieves minimax optimal regret bound (up to logarithmic factors) of O(T^2/3) in the oblivious adversarial setting and a bound of O(min{log(T)/∆²,T^2/3}) in the stochastically-constrained regime, both with (unit) switching costs, where ∆ is the gap between the arms. In the stochastically constrained case, our bound improves over previous results due to [14], that achieved regret of O(T^1/3/∆). We accompany our results with a lower bound showing that, in general, Ω̃(min{1/∆²,T^2/3}) switching cost regret is unavoidable in the stochastically-constrained case for algorithms with O(T^2/3) worst-case switching cost regret.