  The widespread proliferation of data-driven decision-making has ushered in a
recent interest in the design of privacy-preserving algorithms. In this paper,
we consider the ubiquitous problem of gaussian process (GP) bandit optimization
from the lens of privacy-preserving statistics. We propose a solution for
differentially private GP bandit optimization that combines a uniform kernel
approximator with random perturbations, providing a generic framework to create
differentially-private (DP) Gaussian process bandit algorithms. For two
specific DP settings - joint and local differential privacy, we provide
algorithms based on efficient quadrature Fourier feature approximators, that
are computationally efficient and provably no-regret for popular stationary
kernel functions. Our algorithms maintain differential privacy throughout the
optimization procedure and critically do not rely explicitly on the sample path
for prediction, making the parameters straightforward to release as well.
