UCB-based Algorithms for Multinomial Logistic Regression Bandits
This work addresses a practical gap in bandit algorithms for scenarios with more than two user outcomes, such as in online advertising or recommendation systems, though it is incremental as it extends existing UCB methods to a multinomial setting.
The paper tackles the problem of multinomial logistic regression bandits, where a learner aims to maximize expected revenue from a user selecting among multiple outcomes, by proposing MNL-UCB, an algorithm that achieves a regret bound of ̃O(dK√T) with improved dependency on problem-dependent constants.
Out of the rich family of generalized linear bandits, perhaps the most well studied ones are logisitc bandits that are used in problems with binary rewards: for instance, when the learner/agent tries to maximize the profit over a user that can select one of two possible outcomes (e.g., `click' vs `no-click'). Despite remarkable recent progress and improved algorithms for logistic bandits, existing works do not address practical situations where the number of outcomes that can be selected by the user is larger than two (e.g., `click', `show me later', `never show again', `no click'). In this paper, we study such an extension. We use multinomial logit (MNL) to model the probability of each one of $K+1\geq 2$ possible outcomes (+1 stands for the `not click' outcome): we assume that for a learner's action $\mathbf{x}_t$, the user selects one of $K+1\geq 2$ outcomes, say outcome $i$, with a multinomial logit (MNL) probabilistic model with corresponding unknown parameter $\bar{\boldsymbolθ}_{\ast i}$. Each outcome $i$ is also associated with a revenue parameter $ρ_i$ and the goal is to maximize the expected revenue. For this problem, we present MNL-UCB, an upper confidence bound (UCB)-based algorithm, that achieves regret $\tilde{\mathcal{O}}(dK\sqrt{T})$ with small dependency on problem-dependent constants that can otherwise be arbitrarily large and lead to loose regret bounds. We present numerical simulations that corroborate our theoretical results.