  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 ($\epsilon,\delta$)-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
na\"ive 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 ($\epsilon,\delta$)-PAC Pareto set, and the success of identification.
