  This paper studies active learning in the context of robust statistics.
Specifically, we propose a variant of the Best Arm Identification problem for
\emph{contaminated bandits}, where each arm pull has probability $\varepsilon$
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 this distribution. The primary
challenge of the contaminated bandit setting is that the true distributions are
only partially identifiable, even with infinite samples. To address this, we
develop tight, non-asymptotic sample complexity bounds for high-probability
estimation of the first two robust moments (median and median absolute
deviation) from contaminated samples. These concentration inequalities are the
main technical contributions of the paper and may be of independent interest.
Using these results, we adapt several classical Best Arm Identification
algorithms to the contaminated bandit setting and derive sample complexity
upper bounds for our problem. Finally, we provide matching
information-theoretic lower bounds on the sample complexity (up to a small
logarithmic factor).
