Privately Learning Thresholds: Closing the Exponential Gap

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|). Keywords: Differential privacy, Private PAC learning, Halfspaces, Quasi-concave functions