UCB-based Algorithms for Multinomial Logistic Regression Bandits

Out of the rich family of generalized linear bandits, perhaps the most well studied ones are logistic bandits that are used in problems with binary rewards: for instance, when the learner aims 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 MNL probabilistic model with corresponding unknown parameter $\bar{\boldsymbol{\theta}}_{\ast i}$. Each outcome $i$ is also associated with a revenue parameter $\rho_i$ 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.