  This paper analyzes the problem of Gaussian process (GP) bandits with
deterministic observations. The analysis uses a branch and bound algorithm that
is related to the UCB algorithm of (Srinivas et al, 2010). For GPs with
Gaussian observation noise, with variance strictly greater than zero, Srinivas
et al proved that the regret vanishes at the approximate rate of
$O(1/\sqrt{t})$, where t is the number of observations. To complement their
result, we attack the deterministic case and attain a much faster exponential
convergence rate. Under some regularity assumptions, we show that the regret
decreases asymptotically according to $O(e^{-\frac{\tau t}{(\ln t)^{d/4}}})$
with high probability. Here, d is the dimension of the search space and tau is
a constant that depends on the behaviour of the objective function near its
global maximum.
