  In this paper, we study the stochastic linear bandit problem under the
additional requirements of differential privacy, robustness and batched
observations. In particular, we assume an adversary randomly chooses a constant
fraction of the observed rewards in each batch, replacing them with arbitrary
numbers. We present differentially private and robust variants of the arm
elimination algorithm using logarithmic batch queries under two privacy models
and provide regret bounds in both settings. In the first model, every reward in
each round is reported by a potentially different client, which reduces to
standard local differential privacy (LDP). In the second model, every action is
"owned" by a different client, who may aggregate the rewards over multiple
queries and privatize the aggregate response instead. To the best of our
knowledge, our algorithms are the first simultaneously providing differential
privacy and adversarial robustness in the stochastic linear bandits problem.
