Best Arm Identification for Contaminated Bandits

This paper studies active learning in the context of robust statistics. Specifically, we propose the Contaminated Best Arm Identification variant of the multi-armed bandit problem, in which every arm pull has probability ε of generating a sample from an arbitrary contamination distribution instead of the true underlying distribution. The goal is to identify the best (or approximately best) true distribution with high probability, with a secondary goal of providing guarantees on the quality of that arm’s underlying distribution. 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, statistics can be estimated only up to some unavoidable bias. We present tight, non-asymptotic sample complexity bounds for estimating the first two robust moments (median and median absolute deviation) with high probability. We then show how to use this algorithmically for our problem by adapting Best Arm Identification algorithms from the classical multi-armed bandit literature. We give matching upper and lower bounds (up to a small logarithmic factor) on these algorithms’ sample complexities. These results suggest an inherent robustness of classical Best Arm Identification algorithms.