Filtered Poisson Process Bandit on a Continuum
This work addresses a novel bandit problem with filtered Poisson processes, which is incremental as it extends existing continuum-armed bandit frameworks to incorporate stochastic filtering effects.
The paper tackles the problem of maximizing expected rewards in a continuum-armed bandit setting where actions generate filtered Poisson process observations, achieving an O(T^(2/3)) regret bound under Lipschitz assumptions and matching lower bounds up to logarithmic factors.
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.