  We consider optimizing a function network in the noise-free grey-box setting
with RKHS function classes, where the exact intermediate results are
observable. We assume that the structure of the network is known (but not the
underlying functions comprising it), and we study three types of structures:
(1) chain: a cascade of scalar-valued functions, (2) multi-output chain: a
cascade of vector-valued functions, and (3) feed-forward network: a fully
connected feed-forward network of scalar-valued functions. We propose a
sequential upper confidence bound based algorithm GPN-UCB along with a general
theoretical upper bound on the cumulative regret. In addition, we propose a
non-adaptive sampling based method along with its theoretical upper bound on
the simple regret for the Mat\'ern kernel. We also provide
algorithm-independent lower bounds on the simple regret and cumulative regret.
Our regret bounds for GPN-UCB have the same dependence on the time horizon as
the best known in the vanilla black-box setting, as well as near-optimal
dependencies on other parameters (e.g., RKHS norm and network length).
