Abstract
We present a private agnostic learner for halfspaces over an arbitrary finite domain X ⊂ Rd 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|).
Original language | English |
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Pages (from-to) | 269-282 |
Number of pages | 14 |
Journal | Proceedings of Machine Learning Research |
Volume | 99 |
State | Published - 2019 |
Externally published | Yes |
Event | 32nd Conference on Learning Theory, COLT 2019 - Phoenix, United States Duration: 25 Jun 2019 → 28 Jun 2019 |
Funding
Funders | Funder number |
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National Science Foundation | CCF-1412958, 1565387 |
Simons Foundation | |
Israel Science Foundation | 152/17 |
Keywords
- Differential privacy
- Halfspaces
- Private PAC learning
- Quasi-concave functions