  Stochastic linear bandits are a natural and well-studied model for structured
exploration/exploitation problems and are widely used in applications such as
online marketing and recommendation. One of the main challenges faced by
practitioners hoping to apply existing algorithms is that usually the feedback
is randomly delayed and delays are only partially observable. For example,
while a purchase is usually observable some time after the display, the
decision of not buying is never explicitly sent to the system. In other words,
the learner only observes delayed positive events. We formalize this problem as
a novel stochastic delayed linear bandit and propose ${\tt OTFLinUCB}$ and
${\tt OTFLinTS}$, two computationally efficient algorithms able to integrate
new information as it becomes available and to deal with the permanently
censored feedback. We prove optimal $\tilde O(\smash{d\sqrt{T}})$ bounds on the
regret of the first algorithm and study the dependency on delay-dependent
parameters. Our model, assumptions and results are validated by experiments on
simulated and real data.
