  We consider a novel dynamic pricing and learning setting where in addition to
setting prices of products in sequential rounds, the seller also ex-ante
commits to 'advertising schemes'. That is, in the beginning of each round the
seller can decide what kind of signal they will provide to the buyer about the
product's quality upon realization. Using the popular Bayesian persuasion
framework to model the effect of these signals on the buyers' valuation and
purchase responses, we formulate the problem of finding an optimal design of
the advertising scheme along with a pricing scheme that maximizes the seller's
expected revenue. Without any apriori knowledge of the buyers' demand function,
our goal is to design an online algorithm that can use past purchase responses
to adaptively learn the optimal pricing and advertising strategy. We study the
regret of the algorithm when compared to the optimal clairvoyant price and
advertising scheme.
  Our main result is a computationally efficient online algorithm that achieves
an $O(T^{2/3}(m\log T)^{1/3})$ regret bound when the valuation function is
linear in the product quality. Here $m$ is the cardinality of the discrete
product quality domain and $T$ is the time horizon. This result requires some
natural monotonicity and Lipschitz assumptions on the valuation function, but
no Lipschitz or smoothness assumption on the buyers' demand function. For
constant $m$, our result matches the regret lower bound for dynamic pricing
within logarithmic factors, which is a special case of our problem. We also
obtain several improved results for the widely considered special case of
additive valuations, including an $\tilde{O}(T^{2/3})$ regret bound independent
of $m$ when $m\le T^{1/3}$.
