  Scaling Bayesian optimisation (BO) to high-dimensional search spaces is a
active and open research problems particularly when no assumptions are made on
function structure. The main reason is that at each iteration, BO requires to
find global maximisation of acquisition function, which itself is a non-convex
optimization problem in the original search space. With growing dimensions, the
computational budget for this maximisation gets increasingly short leading to
inaccurate solution of the maximisation. This inaccuracy adversely affects both
the convergence and the efficiency of BO. We propose a novel approach where the
acquisition function only requires maximisation on a discrete set of low
dimensional subspaces embedded in the original high-dimensional search space.
Our method is free of any low dimensional structure assumption on the function
unlike many recent high-dimensional BO methods. Optimising acquisition function
in low dimensional subspaces allows our method to obtain accurate solutions
within limited computational budget. We show that in spite of this convenience,
our algorithm remains convergent. In particular, cumulative regret of our
algorithm only grows sub-linearly with the number of iterations. More
importantly, as evident from our regret bounds, our algorithm provides a way to
trade the convergence rate with the number of subspaces used in the
optimisation. Finally, when the number of subspaces is "sufficiently large",
our algorithm's cumulative regret is at most
$\mathcal{O}^{*}(\sqrt{T\gamma_T})$ as opposed to
$\mathcal{O}^{*}(\sqrt{DT\gamma_T})$ for the GP-UCB of Srinivas et al. (2012),
reducing a crucial factor $\sqrt{D}$ where $D$ being the dimensional number of
input space.
