  We consider a version of the continuum armed bandit where an action induces a
filtered realisation of a non-homogeneous Poisson process. Point data in the
filtered sample are then revealed to the decision-maker, whose reward is the
total number of revealed points. Using knowledge of the function governing the
filtering, but without knowledge of the Poisson intensity function, the
decision-maker seeks to maximise the expected number of revealed points over T
rounds. We propose an upper confidence bound algorithm for this problem
utilising data-adaptive discretisation of the action space. This approach
enjoys O(T^(2/3)) regret under a Lipschitz assumption on the reward function.
We provide lower bounds on the regret of any algorithm for the problem, via new
lower bounds for related finite-armed bandits, and show that the orders of the
upper and lower bounds match up to a logarithmic factor.
