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We derive improved regret bounds for the Tsallis-INF algorithm of Zimmert andSeldin (2021). We show that in adversarial regimes with a $(\Delta,C,T)$self-bounding constraint the algorithm achieves$\mathcal{O}\left(\left(\sum_{i\neq i^*}\frac{1}{\Delta_i}\right)\log_+\left(\frac{(K-1)T}{\left(\sum_{i\neq i^*}\frac{1}{\Delta_i}\right)^2}\right)+\sqrt{C\left(\sum_{i\neqi^*}\frac{1}{\Delta_i}\right)\log_+\left(\frac{(K-1)T}{C\sum_{i\neqi^*}\frac{1}{\Delta_i}}\right)}\right)$ regret bound, where $T$ is the timehorizon, $K$ is the number of arms, $\Delta_i$ are the suboptimality gaps,$i^*$ is the best arm, $C$ is the corruption magnitude, and $\log_+(x) =\max\left(1,\log x\right)$. The regime includes stochastic bandits,stochastically constrained adversarial bandits, and stochastic bandits withadversarial corruptions as special cases. Additionally, we provide a generalanalysis, which allows to achieve the same kind of improvement forgeneralizations of Tsallis-INF to other settings beyond multiarmed bandits.

Improved Analysis of the Tsallis-INF Algorithm in Stochastically Constrained Adversarial Bandits and Stochastic Bandits with Adversarial Corruptions