Private Center Points and Learning of Halfspaces

We present a private agnostic learner for halfspaces over an arbitrary finite domain X ⊂ R^d with sample complexity poly(d, 2^{log∗ |}X^|). The building block for this learner is a differentially private algorithm for locating an approximate center point of m > 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 = Ω(d+ log^∗ |X|), whereas for pure differential privacy m = Ω(d log |X|).