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We present a private learner for halfspaces over an arbitrary finite domain$X\subset \mathbb{R}^d$ with sample complexity $mathrm{poly}(d,2^{\log^*|X|})$.The building block for this learner is a differentially private algorithm forlocating an approximate center point of $m>\mathrm{poly}(d,2^{\log^*|X|})$points -- a high dimensional generalization of the median function. Ourconstruction establishes a relationship between these two problems that isreminiscent of the relation between the median and learning one-dimensionalthresholds [Bun et al.\ FOCS '15]. This relationship suggests that the problemof privately locating a center point may have further applications in thedesign of differentially private algorithms.  We also provide a lower bound on the sample complexity for privately findinga point in the convex hull. For approximate differential privacy, we show alower bound of $m=\Omega(d+\log^*|X|)$, whereas for pure differential privacy$m=\Omega(d\log|X|)$.

Private Center Points and Learning of Halfspaces