  We consider the problem of optimizing a vector-valued objective function
$\boldsymbol{f}$ sampled from a Gaussian Process (GP) whose index set is a
well-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 $\boldsymbol{f}(x)$. Since
identifying the Pareto optimal designs via exhaustive search is infeasible when
the cardinality of ${\cal X}$ is large, we propose an algorithm, called
Adaptive $\boldsymbol{\epsilon}$-PAL, that exploits the smoothness of the
GP-sampled function and the structure of $({\cal X},d)$ to learn fast. In
essence, Adaptive $\boldsymbol{\epsilon}$-PAL employs a tree-based adaptive
discretization technique to identify an $\boldsymbol{\epsilon}$-accurate Pareto
set of designs in as few evaluations as possible. We provide both
information-type and metric dimension-type bounds on the sample complexity of
$\boldsymbol{\epsilon}$-accurate Pareto set identification. We also
experimentally show that our algorithm outperforms other Pareto set
identification methods on several benchmark datasets.
