  We consider Bayesian optimization of the output of a network of functions,
where each function takes as input the output of its parent nodes, and where
the network takes significant time to evaluate. Such problems arise, for
example, in reinforcement learning, engineering design, and manufacturing.
While the standard Bayesian optimization approach observes only the final
output, our approach delivers greater query efficiency by leveraging
information that the former ignores: intermediate output within the network.
This is achieved by modeling the nodes of the network using Gaussian processes
and choosing the points to evaluate using, as our acquisition function, the
expected improvement computed with respect to the implied posterior on the
objective. Although the non-Gaussian nature of this posterior prevents
computing our acquisition function in closed form, we show that it can be
efficiently maximized via sample average approximation. In addition, we prove
that our method is asymptotically consistent, meaning that it finds a globally
optimal solution as the number of evaluations grows to infinity, thus
generalizing previously known convergence results for the expected improvement.
Notably, this holds even though our method might not evaluate the domain
densely, instead leveraging problem structure to leave regions unexplored.
Finally, we show that our approach dramatically outperforms standard Bayesian
optimization methods in several synthetic and real-world problems.
