Nearly Minimax Optimal Regret for Multinomial Logistic Bandit
<|reference_start|>Nearly Minimax Optimal Regret for Multinomial Logistic Bandit: In this paper, we study the contextual multinomial logit (MNL) bandit problem in which a learning agent sequentially selects an assortment based on contextual information, and user feedback follows an MNL choice model. There has been a significant discrepancy between lower and upper regret bounds, particularly regarding the maximum assortment size $K$. Additionally, the variation in reward structures between these bounds complicates the quest for optimality. Under uniform rewards, where all items have the same expected reward, we establish a regret lower bound of $\Omega(d\sqrt{\smash[b]{T/K}})$ and propose a constant-time algorithm, OFU-MNL+, that achieves a matching upper bound of $\tilde{O}(d\sqrt{\smash[b]{T/K}})$. Under non-uniform rewards, we prove a lower bound of $\Omega(d\sqrt{T})$ and an upper bound of $\tilde{O}(d\sqrt{T})$, also achievable by OFU-MNL+. Our empirical studies support these theoretical findings. To the best of our knowledge, this is the first work in the contextual MNL bandit literature to prove minimax optimality -- for either uniform or non-uniform reward setting -- and to propose a computationally efficient algorithm that achieves this optimality up to logarithmic factors.<|reference_end|>
@article{lee2024nearly,
    title={Nearly Minimax Optimal Regret for Multinomial Logistic Bandit},
    author={Joongkyu Lee, Min-hwan Oh},
    journal={arXiv preprint arXiv:2405.09831},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.09831},
    primaryClass={stat.ML cs.LG}
}