  Bayesian optimisation is a popular method for efficient optimisation of
expensive black-box functions. Traditionally, BO assumes that the search space
is known. However, in many problems, this assumption does not hold. To this
end, we propose a novel BO algorithm which expands (and shifts) the search
space over iterations based on controlling the expansion rate thought a
hyperharmonic series. Further, we propose another variant of our algorithm that
scales to high dimensions. We show theoretically that for both our algorithms,
the cumulative regret grows at sub-linear rates. Our experiments with synthetic
and real-world optimisation tasks demonstrate the superiority of our algorithms
over the current state-of-the-art methods for Bayesian optimisation in unknown
search space.
