Best of both worlds: Stochastic & adversarial bestarm identification
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Proceedings of the 31st Conference On Learning Theory, PMLR 75:918949, 2018.
Abstract
We study bandit bestarm 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: $\backslash$emph{\{}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 parameterfree 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.
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