  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
$\bar{\boldsymbol\theta}_{\ast i}$. 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.
