  Contextual bandit algorithms are useful in personalized online
decision-making. However, many applications such as personalized medicine and
online advertising require the utilization of individual-specific information
for effective learning, while user's data should remain private from the server
due to privacy concerns. This motivates the introduction of local differential
privacy (LDP), a stringent notion in privacy, to contextual bandits. In this
paper, we design LDP algorithms for stochastic generalized linear bandits to
achieve the same regret bound as in non-privacy settings. Our main idea is to
develop a stochastic gradient-based estimator and update mechanism to ensure
LDP. We then exploit the flexibility of stochastic gradient descent (SGD),
whose theoretical guarantee for bandit problems is rarely explored, in dealing
with generalized linear bandits. We also develop an estimator and update
mechanism based on Ordinary Least Square (OLS) for linear bandits. Finally, we
conduct experiments with both simulation and real-world datasets to demonstrate
the consistently superb performance of our algorithms under LDP constraints
with reasonably small parameters $(\varepsilon, \delta)$ to ensure strong
privacy protection.
