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Many modern statistical applications involve a two-level sampling scheme thatfirst samples subjects from a population and then samples observations on eachsubject. These schemes often are designed to learn both the population-levelfunctional structures shared by the subjects and the functional characteristicsspecific to individual subjects. Common wisdom suggests that learningpopulation-level structures benefits from sampling more subjects whereaslearning subject-specific structures benefits from deeper sampling within eachsubject. Oftentimes these two objectives compete for limited samplingresources, which raises the question of how to optimally sample at the twolevels. We quantify such sampling-depth trade-offs by establishing the $L_2$minimax risk rates for learning the population-level and subject-specificstructures under a hierarchical Gaussian process model framework where weconsider a Bayesian and a frequentist perspective on the unknownpopulation-level structure. These rates provide general lessons for designingtwo-level sampling schemes. Interestingly, subject-specific learningoccasionally benefits more by sampling more subjects than by deeperwithin-subject sampling. We also construct estimators that adapt to unknownsmoothness and achieve the corresponding minimax rates. We conduct twosimulation experiments validating our theory and illustrating the samplingtrade-off in practice, and apply these estimators to two real datasets.

Sampling depth trade-off in function estimation under a two-level design