Constrained regret minimization for multi-criterion multi-armed bandits

We consider a stochastic multi-armed bandit setting and study the problem of regret minimization over a given time horizon, subject to a risk constraint. Each arm is associated with an unknown cost/loss distribution. The learning agent is characterized by a risk-appetite that she is willing to tolerate, which we model using a pre-specified upper bound on the Conditional Value at Risk (CVaR). An optimal arm is one that minimizes the expected loss, among those arms that satisfy the CVaR constraint. The agent is interested in minimizing the number of pulls of suboptimal arms, including the ones that are 'too risky.' For this problem, we propose a Risk-Constrained Lower Confidence Bound (RC-LCB) algorithm, that guarantees logarithmic regret, i.e., the average number of plays of all non-optimal arms is at most logarithmic in the horizon. The algorithm also outputs a boolean flag that correctly identifies with high probability, whether the given instance was feasible/infeasible with respect to the risk constraint. We prove lower bounds on the performance of any risk-constrained regret minimization algorithm and establish a fundamental trade-off between regret minimization and feasibility identification. The proposed algorithm and analyses can be readily generalized to solve constrained multi-criterion optimization problems in the bandits setting.