  Motivated by applications of bandit algorithms in education, we consider a
stochastic multi-armed bandit problem with $\varepsilon$-contaminated rewards.
We allow an adversary to give arbitrary unbounded contaminated rewards with
full knowledge of the past and future. We impose the constraint that for each
time $t$ the proportion of contaminated rewards for any action is less than or
equal to $\varepsilon$. We derive concentration inequalities for two robust
mean estimators for sub-Gaussian distributions in the
$\varepsilon$-contamination context. We define the $\varepsilon$-contaminated
stochastic bandit problem and use our robust mean estimators to give two
variants of a robust Upper Confidence Bound (UCB) algorithm, crUCB. Using
regret derived from only the underlying stochastic rewards, both variants of
crUCB achieve $\mathcal{O} (\sqrt{KT\log T})$ regret for small enough
contamination proportions. Our simulations assume small horizons, reflecting
the newly explored setting of bandits in education. We show that in certain
adversarial regimes crUCB not only outperforms algorithms designed for
stochastic (UCB1) and adversarial (EXP3) bandits but also those that have "best
of both worlds" guarantees (EXP3++ and TsallisInf) even when our constraint on
the proportion of contaminated rewards is broken.
