LC-Tsallis-INF: Generalized Best-of-Both-Worlds Linear Contextual Bandits
<|reference_start|>LC-Tsallis-INF: Generalized Best-of-Both-Worlds Linear Contextual Bandits: This study considers the linear contextual bandit problem with independent and identically distributed (i.i.d.) contexts. In this problem, existing studies have proposed Best-of-Both-Worlds (BoBW) algorithms whose regrets satisfy $O(\log^2(T))$ for the number of rounds $T$ in a stochastic regime with a suboptimality gap lower-bounded by a positive constant, while satisfying $O(\sqrt{T})$ in an adversarial regime. However, the dependency on $T$ has room for improvement, and the suboptimality-gap assumption can be relaxed. For this issue, this study proposes an algorithm whose regret satisfies $O(\log(T))$ in the setting when the suboptimality gap is lower-bounded. Furthermore, we introduce a margin condition, a milder assumption on the suboptimality gap. That condition characterizes the problem difficulty linked to the suboptimality gap using a parameter $\beta \in (0, \infty]$. We then show that the algorithm's regret satisfies $O\left(\left\{\log(T)\right\}^{\frac{1+\beta}{2+\beta}}T^{\frac{1}{2+\beta}}\right)$. Here, $\beta= \infty$ corresponds to the case in the existing studies where a lower bound exists in the suboptimality gap, and our regret satisfies $O(\log(T))$ in that case. Our proposed algorithm is based on the Follow-The-Regularized-Leader with the Tsallis entropy and referred to as the $\alpha$-Linear-Contextual (LC)-Tsallis-INF.<|reference_end|>
@article{kato2024lc-tsallis-inf:,
    title={LC-Tsallis-INF: Generalized Best-of-Both-Worlds Linear Contextual
  Bandits},
    author={Masahiro Kato and Shinji Ito},
    journal={arXiv preprint arXiv:2403.03219},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.03219},
    primaryClass={cs.LG stat.ML}
}