  Bayesian Optimization is a very effective tool for optimizing expensive
black-box functions. Inspired by applications developing and characterizing
reaction chemistry using droplet microfluidic reactors, we consider a novel
setting where the expense of evaluating the function can increase significantly
when making large input changes between iterations. We further assume we are
working asynchronously, meaning we have to select new queries before evaluating
previous experiments. This paper investigates the problem and introduces
'Sequential Bayesian Optimization via Adaptive Connecting Samples' (SnAKe),
which provides a solution by considering large batches of queries and
preemptively building optimization paths that minimize input costs. We
investigate some convergence properties and empirically show that the algorithm
is able to achieve regret similar to classical Bayesian Optimization algorithms
in both synchronous and asynchronous settings, while reducing input costs
significantly. We show the method is robust to the choice of its single
hyper-parameter and provide a parameter-free alternative.
