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.