  Online advertising and product recommendation are important domains of
applications for multi-armed bandit methods. In these fields, the reward that
is immediately available is most often only a proxy for the actual outcome of
interest, which we refer to as a conversion. For instance, in web advertising,
clicks can be observed within a few seconds after an ad display but the
corresponding sale --if any-- will take hours, if not days to happen. This
paper proposes and investigates a new stochas-tic multi-armed bandit model in
the framework proposed by Chapelle (2014) --based on empirical studies in the
field of web advertising-- in which each action may trigger a future reward
that will then happen with a stochas-tic delay. We assume that the probability
of conversion associated with each action is unknown while the distribution of
the conversion delay is known, distinguishing between the (idealized) case
where the conversion events may be observed whatever their delay and the more
realistic setting in which late conversions are censored. We provide
performance lower bounds as well as two simple but efficient algorithms based
on the UCB and KLUCB frameworks. The latter algorithm, which is preferable when
conversion rates are low, is based on a Poissonization argument, of independent
interest in other settings where aggregation of Bernoulli observations with
different success probabilities is required.
