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We are given a set $S$ of $n$ points in $\mathbb{R}^d$ and a (hidden) vector $\mathbf{w}\in\mathbb{R}^d$, where each point $\mathbf{v}\in S$ is associated with a binary label equal to the sign of $\mathbf{w}^{\top}\mathbf{v}$ (assume that $\mathbf{w}^{\top}\mathbf{v}\neq 0$ for all $\mathbf{v}\in S$). Set $S$ is known, but both the labelling assignment of its points and vector $\mathbf{w}$ are initially unknown.

In a sequential fashion, at each time step, we can select one point of $S$ and ask for its label. Then its label is revealed.

Question: What is the minimum number $q$ (expressed as a function of $n$ and $d$) of point label queries necessary to determine the labels of all points in $S$, over all possible sets $S\in\mathbb{R}^d$?

Example: If $d=1$, $q$ is logarithmic in $n$.

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    $\begingroup$ If $d=2$, the answer is already $n$ (consider $n$ points on the circle out of which only one (unknown to you) is on the "other side"). $\endgroup$
    – fedja
    Commented Jul 20, 2019 at 17:27
  • $\begingroup$ OK, so the problem is not interesting. Thank you. $\endgroup$
    – Penelope Benenati
    Commented Jul 20, 2019 at 23:56
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    $\begingroup$ If $S$ forms a grid, then a set of points whose labels reveal the labels of the whole set is called a teaching set. There is a bunch of research on minimal teaching sets. $\endgroup$
    – Max Alekseyev
    Commented Jul 21, 2019 at 13:19
  • $\begingroup$ Thank you Max Alekseyev. $\endgroup$
    – Penelope Benenati
    Commented Jul 21, 2019 at 16:46

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