  In this paper, we introduce the first principled adaptive-sampling procedure
for learning a convex function in the $L_\infty$ norm, a problem that arises
often in the behavioral and social sciences. We present a function-specific
measure of complexity and use it to prove that, for each convex function
$f_{\star}$, our algorithm nearly attains the information-theoretically
optimal, function-specific error rate. We also corroborate our theoretical
contributions with numerical experiments, finding that our method substantially
outperforms passive, uniform sampling for favorable synthetic and data-derived
functions in low-noise settings with large sampling budgets. Our results also
suggest an idealized "oracle strategy", which we use to gauge the potential
advance of any adaptive-sampling strategy over passive sampling, for any given
convex function.
