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We consider the problem of optimizing a vector-valued objective function$\boldsymbol{f}$ sampled from a Gaussian Process (GP) whose index set is awell-behaved, compact metric space $({\cal X},d)$ of designs. We assume that$\boldsymbol{f}$ is not known beforehand and that evaluating $\boldsymbol{f}$at design $x$ results in a noisy observation of $\boldsymbol{f}(x)$. Sinceidentifying the Pareto optimal designs via exhaustive search is infeasible whenthe cardinality of ${\cal X}$ is large, we propose an algorithm, calledAdaptive $\boldsymbol{\epsilon}$-PAL, that exploits the smoothness of theGP-sampled function and the structure of $({\cal X},d)$ to learn fast. Inessence, Adaptive $\boldsymbol{\epsilon}$-PAL employs a tree-based adaptivediscretization technique to identify an $\boldsymbol{\epsilon}$-accurate Paretoset of designs in as few evaluations as possible. We provide bothinformation-type and metric dimension-type bounds on the sample complexity of$\boldsymbol{\epsilon}$-accurate Pareto set identification. We alsoexperimentally show that our algorithm outperforms other Pareto setidentification methods on several benchmark datasets.

Pareto Active Learning with Gaussian Processes and Adaptive Discretization