Better Best of Both Worlds Bounds for Bandits with Switching Costs

TL;DR

This paper introduces a simple, effective algorithm for bandits with switching costs that achieves optimal regret bounds in both adversarial and stochastic regimes, improving upon previous results.

Contribution

The paper presents a novel algorithm that attains minimax optimal regret in adversarial settings and improved bounds in stochastic regimes for bandits with switching costs.

Findings

Achieves $ ilde{O}(T^{2/3})$ regret in adversarial setting.

Attains $ ilde{O}(rac{ ext{log}(T)}{ riangle^2})$ regret in stochastic setting.

Provides a lower bound showing certain regret is unavoidable.

Abstract

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.