  We consider Bayesian optimization of an expensive-to-evaluate black-box
objective function, where we also have access to cheaper approximations of the
objective. In general, such approximations arise in applications such as
reinforcement learning, engineering, and the natural sciences, and are subject
to an inherent, unknown bias. This model discrepancy is caused by an inadequate
internal model that deviates from reality and can vary over the domain, making
the utilization of these approximations a non-trivial task.
  We present a novel algorithm that provides a rigorous mathematical treatment
of the uncertainties arising from model discrepancies and noisy observations.
Its optimization decisions rely on a value of information analysis that extends
the Knowledge Gradient factor to the setting of multiple information sources
that vary in cost: each sampling decision maximizes the predicted benefit per
unit cost.
  We conduct an experimental evaluation that demonstrates that the method
consistently outperforms other state-of-the-art techniques: it finds designs of
considerably higher objective value and additionally inflicts less cost in the
exploration process.
