Improved Regret Bounds of (Multinomial) Logistic Bandits via Regret-to-Confidence-Set Conversion

Junghyun Lee, Se-Young Yun, Kwang-Sung Jun
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:4474-4482, 2024.

Abstract

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 $S \geq \Vert \theta_\star \Vert_2$, 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v238-lee24d, title = {Improved Regret Bounds of (Multinomial) Logistic Bandits via Regret-to-Confidence-Set Conversion}, author = {Lee, Junghyun and Yun, Se-Young and Jun, Kwang-Sung}, booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics}, pages = {4474--4482}, year = {2024}, editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen}, volume = {238}, series = {Proceedings of Machine Learning Research}, month = {02--04 May}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v238/lee24d/lee24d.pdf}, url = {https://proceedings.mlr.press/v238/lee24d.html}, abstract = {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 $S \geq \Vert \theta_\star \Vert_2$, 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.} }
Endnote
%0 Conference Paper %T Improved Regret Bounds of (Multinomial) Logistic Bandits via Regret-to-Confidence-Set Conversion %A Junghyun Lee %A Se-Young Yun %A Kwang-Sung Jun %B Proceedings of The 27th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2024 %E Sanjoy Dasgupta %E Stephan Mandt %E Yingzhen Li %F pmlr-v238-lee24d %I PMLR %P 4474--4482 %U https://proceedings.mlr.press/v238/lee24d.html %V 238 %X 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 $S \geq \Vert \theta_\star \Vert_2$, 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.
APA
Lee, J., Yun, S. & Jun, K.. (2024). Improved Regret Bounds of (Multinomial) Logistic Bandits via Regret-to-Confidence-Set Conversion. Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 238:4474-4482 Available from https://proceedings.mlr.press/v238/lee24d.html.

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