Constrained regret minimization for multi-criterion multi-armed bandits

We consider a stochastic multi-armed bandit setting and study the problem of constrained regret minimization over given time horizon. Each arm is associated with an unknown, possibly multi-dimensional distribution, merit determined by several, conflicting attributes. The aim to optimize ‘primary’ attribute subject user-provided constraints on other ‘secondary’ assume that attributes can be estimated using samples from arms’ distributions, estimators enjoy suitable concentration properties. propose algorithm called Con-LCB guarantees logarithmic regret, i.e., average number plays all non-optimal arms at most in also outputs boolean flag correctly identifies, high probability, whether instance feasible/infeasible respect constraints. show optimal within universal constant, more sophisticated algorithms cannot do much better universally. Finally, we establish fundamental trade-off between feasibility identification. Our framework finds natural applications, for instance, financial portfolio optimization, where risk maximization expected return meaningful.