Vector Optimization with Stochastic Bandit Feedback
This work addresses vector optimization problems for decision-making under uncertainty, but it is incremental as it extends existing bandit frameworks to polyhedral ordering cones.
The paper tackles the problem of identifying Pareto sets in vector optimization with stochastic bandit feedback, generalizing best arm identification to vector-valued rewards, and shows that the sample complexity scales with the square of a new ordering complexity measure, with experiments verifying these theoretical results.
We introduce vector optimization problems with stochastic bandit feedback, in which preferences among designs are encoded by a polyhedral ordering cone $C$. Our setup generalizes the best arm identification problem to vector-valued rewards by extending the concept of Pareto set beyond multi-objective optimization. We characterize the sample complexity of ($ε,δ$)-PAC Pareto set identification by defining a new cone-dependent notion of complexity, called the ordering complexity. In particular, we provide gap-dependent and worst-case lower bounds on the sample complexity and show that, in the worst-case, the sample complexity scales with the square of ordering complexity. Furthermore, we investigate the sample complexity of the naïve elimination algorithm and prove that it nearly matches the worst-case sample complexity. Finally, we run experiments to verify our theoretical results and illustrate how $C$ and sampling budget affect the Pareto set, the returned ($ε,δ$)-PAC Pareto set, and the success of identification.