  How can we efficiently gather information to optimize an unknown function,
when presented with multiple, mutually dependent information sources with
different costs? For example, when optimizing a robotic system, intelligently
trading off computer simulations and real robot testings can lead to
significant savings. Existing methods, such as multi-fidelity GP-UCB or Entropy
Search-based approaches, either make simplistic assumptions on the interaction
among different fidelities or use simple heuristics that lack theoretical
guarantees. In this paper, we study multi-fidelity Bayesian optimization with
complex structural dependencies among multiple outputs, and propose
MF-MI-Greedy, a principled algorithmic framework for addressing this problem.
In particular, we model different fidelities using additive Gaussian processes
based on shared latent structures with the target function. Then we use
cost-sensitive mutual information gain for efficient Bayesian global
optimization. We propose a simple notion of regret which incorporates the cost
of different fidelities, and prove that MF-MI-Greedy achieves low regret. We
demonstrate the strong empirical performance of our algorithm on both synthetic
and real-world datasets.
