  Kernel-based bandit is an extensively studied black-box optimization problem,
in which the objective function is assumed to live in a known reproducing
kernel Hilbert space. While nearly optimal regret bounds (up to logarithmic
factors) are established in the noisy setting, surprisingly, less is known
about the noise-free setting (when the exact values of the underlying function
is accessible without observation noise). We discuss several upper bounds on
regret; none of which seem order optimal, and provide a conjecture on the order
optimal regret bound.
