  Bayesian persuasion studies how an informed sender should influence beliefs
of rational receivers who take decisions through Bayesian updating of a common
prior. We focus on the online Bayesian persuasion framework, in which the
sender repeatedly faces one or more receivers with unknown and adversarially
selected types. First, we show how to obtain a tight $\tilde O(T^{1/2})$ regret
bound in the case in which the sender faces a single receiver and has partial
feedback, improving over the best previously known bound of $\tilde
O(T^{4/5})$. Then, we provide the first no-regret guarantees for the
multi-receiver setting under partial feedback. Finally, we show how to design
no-regret algorithms with polynomial per-iteration running time by exploiting
type reporting, thereby circumventing known intractability results on online
Bayesian persuasion. We provide efficient algorithms guaranteeing a
$O(T^{1/2})$ regret upper bound both in the single- and multi-receiver scenario
when type reporting is allowed.
