LC-Tsalis-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(log²(T)) for the number of rounds T in a stochastic regime with a suboptimality gap lower-bounded by a positive constant, while satisfying O(√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 β ∈ (0, ∞]. We then show that the algorithm's regret satisfies O(log(T)\1+β/2+βT1/2+β). Here, β= ∞ 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 α-Linear-Contextual (LC)-Tsallis-INF.