  Based on differential privacy (DP) framework, we introduce and unify privacy
definitions for the multi-armed bandit algorithms. We represent the framework
with a unified graphical model and use it to connect privacy definitions. We
derive and contrast lower bounds on the regret of bandit algorithms satisfying
these definitions. We leverage a unified proving technique to achieve all the
lower bounds. We show that for all of them, the learner's regret is increased
by a multiplicative factor dependent on the privacy level $\epsilon$. We
observe that the dependency is weaker when we do not require local differential
privacy for the rewards.
