  Bayesian optimization is an effective method for finding extrema of a
black-box function. We propose a new type of Bayesian optimization for learning
user preferences in high-dimensional spaces. The central assumption is that the
underlying objective function cannot be evaluated directly, but instead a
minimizer along a projection can be queried, which we call a projective
preferential query. The form of the query allows for feedback that is natural
for a human to give, and which enables interaction. This is demonstrated in a
user experiment in which the user feedback comes in the form of optimal
position and orientation of a molecule adsorbing to a surface. We demonstrate
that our framework is able to find a global minimum of a high-dimensional
black-box function, which is an infeasible task for existing preferential
Bayesian optimization frameworks that are based on pairwise comparisons.
