  Bayesian Optimisation (BO) methods seek to find global optima of objective
functions which are only available as a black-box or are expensive to evaluate.
Such methods construct a surrogate model for the objective function,
quantifying the uncertainty in that surrogate through Bayesian inference.
Objective evaluations are sequentially determined by maximising an acquisition
function at each step. However, this ancilliary optimisation problem can be
highly non-trivial to solve, due to the non-convexity of the acquisition
function, particularly in the case of batch Bayesian optimisation, where
multiple points are selected in every step. In this work we reformulate batch
BO as an optimisation problem over the space of probability measures. We
construct a new acquisition function based on multipoint expected improvement
which is convex over the space of probability measures. Practical schemes for
solving this `inner' optimisation problem arise naturally as gradient flows of
this objective function. We demonstrate the efficacy of this new method on
different benchmark functions and compare with state-of-the-art batch BO
methods.
