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We consider a constrained, pure exploration, stochastic multi-armed banditformulation under a fixed budget. Each arm is associated with an unknown,possibly multi-dimensional distribution and is described by multiple attributesthat are a function of this distribution. The aim is to optimize a particularattribute subject to user-defined constraints on the other attributes. Thisframework models applications such as financial portfolio optimization, whereit is natural to perform risk-constrained maximization of mean return. Weassume that the attributes can be estimated using samples from the arms'distributions and that these estimators satisfy suitable concentrationinequalities. We propose an algorithm called \textsc{Constrained-SR} based onthe Successive Rejects framework, which recommends an optimal arm and flags theinstance as being feasible or infeasible. A key feature of this algorithm isthat it is designed on the basis of an information theoretic lower bound fortwo-armed instances. We characterize an instance-dependent upper bound on theprobability of error under \textsc{Constrained-SR}, that decays exponentiallywith respect to the budget. We further show that the associated decay rate isnearly optimal relative to an information theoretic lower bound in certainspecial cases.

Constrained Pure Exploration Multi-Armed Bandits with a Fixed Budget