When Privacy Meets Partial Information: A Refined Analysis of Differentially Private Bandits

We study the problem of multi-armed bandits with ε-global Differential Privacy (DP). First, we prove the minimax and problem-dependent regret lower bounds for stochastic and linear bandits that quantify the hardness of bandits with ε-global DP. These bounds suggest the existence of two hardness regimes depending on the privacy budget ε. In the high-privacy regime (small ε), the hardness depends on a coupled effect of privacy and partial information about the reward distributions. In the low-privacy regime (large ε), bandits with ε-global DP are not harder than the bandits without privacy.