  We study the problem of estimating the value of a known smooth function $f$
at an unknown point $\boldsymbol{\mu} \in \mathbb{R}^n$, where each component
$\mu_i$ can be sampled via a noisy oracle. Sampling more frequently components
of $\boldsymbol{\mu}$ corresponding to directions of the function with larger
directional derivatives is more sample-efficient. However, as
$\boldsymbol{\mu}$ is unknown, the optimal sampling frequencies are also
unknown. We design an instance-adaptive algorithm that learns to sample
according to the importance of each coordinate, and with probability at least
$1-\delta$ returns an $\epsilon$ accurate estimate of $f(\boldsymbol{\mu})$. We
generalize our algorithm to adapt to heteroskedastic noise, and prove
asymptotic optimality when $f$ is linear. We corroborate our theoretical
results with numerical experiments, showing the dramatic gains afforded by
adaptivity.
