Best Arm Identification for Contaminated Bandits
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We propose the Contaminated Best Arm Identification variant of the Multi-Armed Bandit problem, in which every arm pull has some probability ε of generating a sample from an arbitrary contamination distribution instead of the true underlying distribution. We would still like to guarantee that we can identify the best (or approximately best) true distribution with high probability, as well as provide guarantees on how good that arm's underlying distribution is. It is simple to see that in this contamination model, there are no consistent estimators for statistics (e.g. median) of the underlying distribution, and that even with infinite samples they can be estimated only up to some unavoidable bias. We give novel tight, finite-sample complexity bounds for estimating the first two robust moments (median and median absolute deviation) with high probability. We then show how to use these algorithmically for our problem by adapting Best Arm Identification algorithms from the classical Multi-Armed Bandit literature. We present matching upper and lower bounds (up to a small logarithmic factor) on these algorithm's sample complexity. These results suggest an inherent robustness of classical Best Arm Identification algorithm.
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