Improved Analysis of Robustness of the Tsallis-INF Algorithm to Adversarial Corruptions in Stochastic Multiarmed Bandits

We derive improved regret bounds for the Tsallis-INF algorithm of Zimmert and Seldin (2021). In the adversarial regime with a self-bounding constraint and the stochastic regime with adversarial corruptions as its special case we improve the dependence on corruption magnitude $C$. In particular, for $C = \Theta\left(\frac{T}{\log T}\right)$, where $T$ is the time horizon, we achieve an improvement by a multiplicative factor of $\sqrt{\frac{\log T}{\log\log T}}$ relative to the bound of Zimmert and Seldin (2021). We also improve the dependence of the regret bound on time horizon from $\log T$ to $\log \frac{(K-1)T}{(\sum_{i\neq i^*}\frac{1}{\Delta_i})^2}$, where $K$ is the number of arms, $\Delta_i$ are suboptimality gaps for suboptimal arms $i$, and $i^*$ is the optimal arm. Additionally, we provide a general analysis, which allows to achieve the same kind of improvement for generalizations of Tsallis-INF to other settings beyond multiarmed bandits.