NDLI: Differentially Private Release and Learning of Threshold Functions

Content Provider	IEEE Xplore Digital Library
Author	Bun, M. Nissim, K. Stemmer, U. Vadhan, S.
Copyright Year	2015
Abstract	We prove new upper and lower bounds on the sample complexity of (ε, δ) differentially private algorithms for releasing approximate answers to threshold functions. A threshold function c over a totally ordered domain X evaluates to $c_{z}(y)$ = 1 if y ≤ x, and evaluates to 0 otherwise. We give the first nontrivial lower bound for releasing thresholds with (ε, δ) differential privacy, showing that the task is impossible over an infinite domain X, and moreover requires sample complexity n ≥ Ω(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+ο(1)) log* \|X\| samples. This improves the previous best upper bound of 8(1+ο(1)) log* \|X\| (Beimel et al., RANDOM '13). Our sample complexity upper and lower bounds also apply to the tasks of learning distributions 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 (ε, δ) differential privacy and learning without privacy. For properly learning thresholds in ℓ dimensions, this lower bound extends to n ≥ Ω(ℓ · 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.
Starting Page	634
Ending Page	649
File Size	265949
Page Count	16
File Format	PDF
ISBN	9781467381918
ISSN	02725428
DOI	10.1109/FOCS.2015.45
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2015-10-17
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Privacy Complexity theory Data privacy Approximation algorithms Upper bound Databases Computer science fingerprinting codes differential privacy PAC learning lower bounds threshold functions
Content Type	Text
Resource Type	Article

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