Logistic bandit is a ubiquitous framework of modeling users' choices, e.g., click vs. no click for advertisement recommender system. We observe that the prior works overlook or neglect dependencies in $S \geq \lVert \theta_\star \rVert_2$, where $\theta_\star \in \mathbb{R}^d$ is the unknown parameter vector, which is particularly problematic when $S$ is large, e.g., $S \geq d$. In this work, we improve the dependency on $S$ via a novel approach called {\it regret-to-confidence set conversion (R2CS)}, which allows us to construct a convex confidence set based on only the \textit{existence} of an online learning algorithm with a regret guarantee. Using R2CS, we obtain a strict improvement in the regret bound w.r.t. $S$ in logistic bandits while retaining computational feasibility and the dependence on other factors such as $d$ and $T$. We apply our new confidence set to the regret analyses of logistic bandits with a new martingale concentration step that circumvents an additional factor of $S$. We then extend this analysis to multinomial logistic bandits and obtain similar improvements in the regret, showing the efficacy of R2CS. While we applied R2CS to the (multinomial) logistic model, R2CS is a generic approach for developing confidence sets that can be used for various models, which can be of independent interest.

通过一种新的方法称为遗憾与置信集转化（R2CS），我们在逻辑回归赌博机中获得了对S的严格遗憾界限的改进，同时保持了计算可行性和对其他因素（如d和T）的依赖。通过引入新的鞅集中步骤来将其应用于多项式逻辑回归赌博机，我们也获得了类似的遗憾改进，展示了R2CS的有效性。

通过遗憾-置信区间转化改进（多项式）逻辑回归赌博机的遗憾界限