Nearly Minimax Optimal Regret for Multinomial Logistic Bandit

• URL: http://arxiv.org/abs/2405.09831v6
• Date: Sun, 03 Nov 2024 00:58:36 GMT
• Title: Nearly Minimax Optimal Regret for Multinomial Logistic Bandit
• Authors: Joongkyu Lee, Min-hwan Oh,
• Abstract summary: We study the contextual multinomial logit (MNL) bandit problem in which a learning agent sequentially selects an assortment based on contextual information. There has been a significant discrepancy between lower and upper regret bounds, particularly regarding the maximum assortment size $K$. We propose a constant-time algorithm, OFU-MNL+, that achieves a matching upper bound of $tildeO(dsqrtsmash[b]T/K)$.
• Score: 8.087699764574788
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• Abstract: 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 $\tilde{O}(d\sqrt{\smash[b]{T/K}})$. We also provide instance-dependent minimax regret bounds under uniform rewards. 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.

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