  This paper analyses 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., 2010) proved that the regret vanishes at the approximate rate of
$O(\frac{1}{\sqrt{t}})$, 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.
