Regret Bounds for Noise-Free Cascaded Kernelized Bandits

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. For the Mat\'ern kernel, we additionally propose a non-adaptive sampling based method along with its theoretical upper bound on the simple regret. We also provide algorithm-independent lower bounds on the simple regret and cumulative regret, showing that GPN-UCB is near-optimal for chains and multi-output chains in broad cases of interest.