  In this work, we study sequential choice bandits with feedback. We propose
bandit algorithms for a platform that personalizes users' experience to
maximize its rewards. For each action directed to a given user, the platform is
given a positive reward, which is a non-decreasing function of the action, if
this action is below the user's threshold. Users are equipped with a patience
budget, and actions that are above the threshold decrease the user's patience.
When all patience is lost, the user abandons the platform. The platform
attempts to learn the thresholds of the users in order to maximize its rewards,
based on two different feedback models describing the information pattern
available to the platform at each action. We define a notion of regret by
determining the best action to be taken when the platform knows that the user's
threshold is in a given interval. We then propose bandit algorithms for the two
feedback models and show that upper and lower bounds on the regret are of the
order of $\tilde{O}(N^{2/3})$ and $\tilde\Omega(N^{2/3})$, respectively, where
$N$ is the total number of users. Finally, we show that the waiting time of any
user before receiving a personalized experience is uniform in $N$.
