  We introduce a new model of stochastic bandits with adversarial corruptions
which aims to capture settings where most of the input follows a stochastic
pattern but some fraction of it can be adversarially changed to trick the
algorithm, e.g., click fraud, fake reviews and email spam. The goal of this
model is to encourage the design of bandit algorithms that (i) work well in
mixed adversarial and stochastic models, and (ii) whose performance
deteriorates gracefully as we move from fully stochastic to fully adversarial
models.
  In our model, the rewards for all arms are initially drawn from a
distribution and are then altered by an adaptive adversary. We provide a simple
algorithm whose performance gracefully degrades with the total corruption the
adversary injected in the data, measured by the sum across rounds of the
biggest alteration the adversary made in the data in that round; this total
corruption is denoted by $C$. Our algorithm provides a guarantee that retains
the optimal guarantee (up to a logarithmic term) if the input is stochastic and
whose performance degrades linearly to the amount of corruption $C$, while
crucially being agnostic to it. We also provide a lower bound showing that this
linear degradation is necessary if the algorithm achieves optimal performance
in the stochastic setting (the lower bound works even for a known amount of
corruption, a special case in which our algorithm achieves optimal performance
without the extra logarithm).
