Differentially Private Release and Learning of Threshold Functions

Mark Bun, Kobbi Nissim, Uri Stemmer, Salil Vadhan

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


We prove new upper and lower bounds on the sample complexity of (∈, δ) differentially private algorithms for releasing approximate answers to threshold functions. A threshold function cx over a totally ordered domain X evaluates to cx(y) = 1 if y ≤ x, and evaluates to 0 otherwise. We give the first nontrivial lower bound for releasing thresholds with (∈, δ) differential privacy, showing that the task is impossible over an infinite domain X, and moreover requires sample complexity n ≥ (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 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 (∈, δ) differential privacy and learning without privacy. For properly learning thresholds in '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.

Original languageEnglish
Title of host publicationProceedings - 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015
PublisherIEEE Computer Society
Number of pages16
ISBN (Electronic)9781467381918
StatePublished - 11 Dec 2015
Event56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015 - Berkeley, United States
Duration: 17 Oct 201520 Oct 2015

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS


Conference56th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2015
Country/TerritoryUnited States


  • PAC learning
  • differential privacy
  • fingerprinting codes
  • lower bounds
  • threshold functions

All Science Journal Classification (ASJC) codes

  • General Computer Science

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