  Learning functions with high-dimensional outputs is critical in many
applications, such as physical simulation and engineering design. However,
collecting training examples for these applications is often costly, e.g. by
running numerical solvers. The recent work (Li et al., 2022) proposes the first
multi-fidelity active learning approach for high-dimensional outputs, which can
acquire examples at different fidelities to reduce the cost while improving the
learning performance. However, this method only queries at one pair of fidelity
and input at a time, and hence has a risk to bring in strongly correlated
examples to reduce the learning efficiency. In this paper, we propose Batch
Multi-Fidelity Active Learning with Budget Constraints (BMFAL-BC), which can
promote the diversity of training examples to improve the benefit-cost ratio,
while respecting a given budget constraint for batch queries. Hence, our method
can be more practically useful. Specifically, we propose a novel batch
acquisition function that measures the mutual information between a batch of
multi-fidelity queries and the target function, so as to penalize highly
correlated queries and encourages diversity. The optimization of the batch
acquisition function is challenging in that it involves a combinatorial search
over many fidelities while subject to the budget constraint. To address this
challenge, we develop a weighted greedy algorithm that can sequentially
identify each (fidelity, input) pair, while achieving a near $(1 -
1/e)$-approximation of the optimum. We show the advantage of our method in
several computational physics and engineering applications.
