Private Center Points and Learning of Halfspaces

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Abstract

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 for locating an approximate center point of$m>\mathrm{poly}(d,2^{\log^|X|})$points – a high dimensional generalization of the median function. Our construction establishes a relationship between these two problems that is reminiscent of the relation between the median and learning one-dimensional thresholds [Bun et al.\ FOCS ‘15]. This relationship suggests that the problem of privately locating a center point may have further applications in the design of differentially private algorithms. We also provide a lower bound on the sample complexity for privately finding a point in the convex hull. For approximate differential privacy, we show a lower bound of$m=\Omega(d+\log^*|X|)$, whereas for pure differential privacy$m=\Omega(d\log|X|)$.

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URL

http://arxiv.org/abs/1902.10731

PDF

http://arxiv.org/pdf/1902.10731