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

Batch Multi-Fidelity Active Learning with Budget Constraints