# Differentially private release and learning of threshold functions

### Citation:

Bun, Mark, Kobbi Nissim, Uri Stemmer, and Salil Vadhan. “Differentially private release and learning of threshold functions.” In Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS ‘15). IEEE, 2015.
 ArXiv2015.pdf 638 KB IEEE2015.pdf 261 KB

### Abstract:

Version HistoryFull version posted as arXiv:1504.07553.

We prove new upper and lower bounds on the sample complexity of $$(\varepsilon, \delta)$$ differentially private algorithms for releasing approximate answers to threshold functions. A threshold function $$c_x$$ over a totally ordered domain $$X$$ evaluates to $$c_x(y)=1$$ if $$y \leq {x}$$, and evaluates to $$0$$ otherwise. We give the first nontrivial lower bound for releasing thresholds with $$(\varepsilon, \delta)$$ differential privacy, showing that the task is impossible over an infinite domain $$X$$, and moreover requires sample complexity $$n \geq \Omega(\log^* |X|)$$, which grows with the size of the domain. Inspired by the techniques used to prove this lower bound, we give an algorithm for releasing thresholds with $$n ≤ 2^{(1+o(1)) \log^∗|X|}$$ samples. This improves the previous best upper bound of $$8^{(1+o(1)) \log^∗ |X|}$$(Beimel et al., RANDOM ’13).

Our sample complexity upper and lower bounds also apply to the tasks of learning distri- butions with respect to Kolmogorov distance and of properly PAC learning thresholds with differential privacy. The lower bound gives the first separation between the sample complexity of properly learning a concept class with $$(\varepsilon, \delta)$$ differential privacy and learning without privacy. For properly learning thresholds in $$\ell$$ dimensions, this lower bound extends to $$n ≥ Ω(\ell·\log^∗ |X|)$$.

To obtain our results, we give reductions in both directions from releasing and properly learning thresholds and the simpler interior point problem. Given a database $$D$$ of elements from $$X$$, the interior point problem asks for an element between the smallest and largest elements in $$D$$. We introduce new recursive constructions for bounding the sample complexity of the interior point problem, as well as further reductions and techniques for proving impossibility results for other basic problems in differential privacy.

Publisher's Version

Last updated on 06/23/2020