Improved Regret Bounds of (Multinomial) Logistic Bandits via Regret-to-Confidence-Set Conversion
This work provides improved theoretical guarantees for logistic bandits, which are widely used in applications like recommendation systems, though it is an incremental improvement focused on a specific technical bottleneck.
The paper tackles the problem of large parameter dependencies in logistic bandit models, which degrade regret bounds when the parameter norm S is large. It introduces a regret-to-confidence set conversion (R2CS) method to improve regret bounds with respect to S while maintaining computational feasibility and dependencies on other factors like dimension d and time horizon T.
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 $S \geq \lVert θ_\star \rVert_2$, where $θ_\star \in \mathbb{R}^d$ 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.