  Bayesian Optimisation (BO) is a technique used in optimising a
$D$-dimensional function which is typically expensive to evaluate. While there
have been many successes for BO in low dimensions, scaling it to high
dimensions has been notoriously difficult. Existing literature on the topic are
under very restrictive settings. In this paper, we identify two key challenges
in this endeavour. We tackle these challenges by assuming an additive structure
for the function. This setting is substantially more expressive and contains a
richer class of functions than previous work. We prove that, for additive
functions the regret has only linear dependence on $D$ even though the function
depends on all $D$ dimensions. We also demonstrate several other statistical
and computational benefits in our framework. Via synthetic examples, a
scientific simulation and a face detection problem we demonstrate that our
method outperforms naive BO on additive functions and on several examples where
the function is not additive.
