Privately Learning Thresholds: Closing the Exponential Gap

Haim Kaplan, Katrina Ligett, Yishay Mansour, Moni Naor, Uri Stemmer

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We present a private agnostic learner for halfspaces over an arbitrary finite domain X ⊂ R d with sample complexity poly(d, 2log∗|X|). The building block for this learner is a differentially private algorithm for locating an approximate center point of m > poly(d, 2log∗|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
Original languageEnglish GB
Title of host publicationConference on Learning Theory, COLT 2020, 9-12 July 2020, Virtual Event [Graz, Austria]
EditorsJacob D. Abernethy, Shivani Agarwal
Number of pages23
StatePublished - 2020

Publication series

NameProceedings of Machine Learning Research

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