  Bayesian optimization (BO) is a powerful approach for optimizing black-box,
expensive-to-evaluate functions. To enable a flexible trade-off between the
cost and accuracy, many applications allow the function to be evaluated at
different fidelities. In order to reduce the optimization cost while maximizing
the benefit-cost ratio, in this paper, we propose Batch Multi-fidelity Bayesian
Optimization with Deep Auto-Regressive Networks (BMBO-DARN). We use a set of
Bayesian neural networks to construct a fully auto-regressive model, which is
expressive enough to capture strong yet complex relationships across all the
fidelities, so as to improve the surrogate learning and optimization
performance. Furthermore, to enhance the quality and diversity of queries, we
develop a simple yet efficient batch querying method, without any combinatorial
search over the fidelities. We propose a batch acquisition function based on
Max-value Entropy Search (MES) principle, which penalizes highly correlated
queries and encourages diversity. We use posterior samples and moment matching
to fulfill efficient computation of the acquisition function and conduct
alternating optimization over every fidelity-input pair, which guarantees an
improvement at each step. We demonstrate the advantage of our approach on four
real-world hyperparameter optimization applications.
