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This paper proposes a linear bandit algorithm that is adaptive toenvironments at two different levels of hierarchy. At the higher level, theproposed algorithm adapts to a variety of types of environments. Moreprecisely, it achieves best-of-three-worlds regret bounds, i.e., of${O}(\sqrt{T \log T})$ for adversarial environments and of $O(\frac{\logT}{\Delta_{\min}} + \sqrt{\frac{C \log T}{\Delta_{\min}}})$ for stochasticenvironments with adversarial corruptions, where $T$, $\Delta_{\min}$, and $C$denote, respectively, the time horizon, the minimum sub-optimality gap, and thetotal amount of the corruption. Note that polynomial factors in thedimensionality are omitted here. At the lower level, in each of the adversarialand stochastic regimes, the proposed algorithm adapts to certain environmentalcharacteristics, thereby performing better. The proposed algorithm hasdata-dependent regret bounds that depend on all of the cumulative loss for theoptimal action, the total quadratic variation, and the path-length of the lossvector sequence. In addition, for stochastic environments, the proposedalgorithm has a variance-adaptive regret bound of $O(\frac{\sigma^2 \logT}{\Delta_{\min}})$ as well, where $\sigma^2$ denotes the maximum variance ofthe feedback loss. The proposed algorithm is based on the SCRiBLe algorithm. Byincorporating into this a new technique we call scaled-up sampling, we obtainhigh-level adaptability, and by incorporating the technique of optimisticonline learning, we obtain low-level adaptability.

Best-of-Three-Worlds Linear Bandit Algorithm with Variance-Adaptive Regret Bounds