We study bandit best-arm identification with arbitrary and potentially adversarial rewards. A simple random uniform learner obtains the optimal rate of error in the adversarial scenario. However, this type of strategy is suboptimal when the rewards are sampled stochastically. Therefore, we ask: Can we design a learner that performs optimally in both the stochastic and adversarial problems while not being aware of the nature of the rewards? First, we show that designing such a learner is impossible in general. In particular, to be robust to adversarial rewards, we can only guarantee optimal rates of error on a subset of the stochastic problems. We give a lower bound that characterizes the optimal rate in stochastic problems if the strategy is constrained to be robust to adversarial rewards. Finally, we design a simple parameter-free algorithm and show that its probability of error matches (up to log factors) the lower bound in stochastic problems, and it is also robust to adversarial ones.

We consider the problem of finding the best arm in a stochastic Mutli-armed Bandit (MAB) game and propose a general framework based on verification that applies to multiple well-motivated generalizations of the classic MAB problem. In these generalizations, additional structure is known in advance, causing the task of verifying the optimality of a candidate to be easier than discovering the best arm. Our results are focused on the scenario where the failure probability $\delta$ must be very low; we essentially show that in this high confidence regime, identifying the best arm is as easy as the task of verification. We demonstrate the effectiveness of our framework by applying it, and improving the state-of-the art results in the problems of: Linear bandits, Dueling bandits with the Condorcet assumption, Copeland dueling bandits, Unimodal bandits and Graphical bandits.

When n is independent ofδ our approach yields an algorithm whose sample complexityconvergesto 2n2 log 1δ asngrows.

The softmax function is ubiquitous in machine learning and optimization applications.

Our analysis highlights sufficient properties for a regret minimization algorithm to be used as leader.

The knowledge gradient (KG) algorithm is a popular policy for the best arm identification (BAI) problem. It is built on the simple idea of always choosing the measurement that yields the greatest expected one-step improvement in the estimate of the best mean of the arms.

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