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Classical multi-armed bandit problems use the expected value of an arm as ametric to evaluate its goodness. However, the expected value is a risk-neutralmetric. In many applications like finance, one is interested in balancing theexpected return of an arm (or portfolio) with the risk associated with thatreturn. In this paper, we consider the problem of selecting the arm thatoptimizes a linear combination of the expected reward and the associatedConditional Value at Risk (CVaR) in a fixed budget best-arm identificationframework. We allow the reward distributions to be unbounded or evenheavy-tailed. For this problem, our goal is to devise algorithms that areentirely distribution oblivious, i.e., the algorithm is not aware of anyinformation on the reward distributions, including bounds on the moments/tails,or the suboptimality gaps across arms.  In this paper, we provide a class of such algorithms with provable upperbounds on the probability of incorrect identification. In the process, wedevelop a novel estimator for the CVaR of unbounded (including heavy-tailed)random variables and prove a concentration inequality for the same, which couldbe of independent interest. We also compare the error bounds for ourdistribution oblivious algorithms with those corresponding to standardnon-oblivious algorithms. Finally, numerical experiments reveal that ouralgorithms perform competitively when compared with non-oblivious algorithms,suggesting that distribution obliviousness can be realised in practice withoutincurring a significant loss of performance.

Distribution oblivious, risk-aware algorithms for multi-armed bandits with unbounded rewards