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Profile Reconstruction from Private Sketches

Hao Wu, Rasmus Pagh

TL;DR

An information-theoretic lower bound is given showing that this dependence on d is asymptotically optimal among all private, updatable sketches for the profile reconstruction problem with a high-probability error guarantee.

Abstract

Given a multiset of $n$ items from $\mathcal{D}$, the \emph{profile reconstruction} problem is to estimate, for $t = 0, 1, \dots, n$, the fraction $\vec{f}[t]$ of items in $\mathcal{D}$ that appear exactly $t$ times. We consider differentially private profile estimation in a distributed, space-constrained setting where we wish to maintain an updatable, private sketch of the multiset that allows us to compute an approximation of $\vec{f} = (\vec{f}[0], \dots, \vec{f}[n])$. Using a histogram privatized using discrete Laplace noise, we show how to ``reverse'' the noise, using an approach of Dwork et al.~(ITCS '10). We show how to speed up their LP-based technique from polynomial time to $O(d + n \log n)$, where $d = |\mathcal{D}|$, and analyze the achievable error in the $\ell_1$, $\ell_2$ and $\ell_\infty$ norms. In all cases the dependency of the error on $d$ is $O( 1 / \sqrt{d})$ -- we give an information-theoretic lower bound showing that this dependence on $d$ is asymptotically optimal among all private, updatable sketches for the profile reconstruction problem with a high-probability error guarantee.

Profile Reconstruction from Private Sketches

TL;DR

An information-theoretic lower bound is given showing that this dependence on d is asymptotically optimal among all private, updatable sketches for the profile reconstruction problem with a high-probability error guarantee.

Abstract

Given a multiset of items from , the \emph{profile reconstruction} problem is to estimate, for , the fraction of items in that appear exactly times. We consider differentially private profile estimation in a distributed, space-constrained setting where we wish to maintain an updatable, private sketch of the multiset that allows us to compute an approximation of . Using a histogram privatized using discrete Laplace noise, we show how to ``reverse'' the noise, using an approach of Dwork et al.~(ITCS '10). We show how to speed up their LP-based technique from polynomial time to , where , and analyze the achievable error in the , and norms. In all cases the dependency of the error on is -- we give an information-theoretic lower bound showing that this dependence on is asymptotically optimal among all private, updatable sketches for the profile reconstruction problem with a high-probability error guarantee.
Paper Structure (28 sections, 15 theorems, 104 equations, 5 algorithms)

This paper contains 28 sections, 15 theorems, 104 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Technical overview
  4. Problem Description
  5. Discrete Laplace Mechanism.
  6. Preliminaries
  7. Near-linear Time Algorithm
  8. Unfolding Clipped Histograms
  9. Searching for Profiles
  10. Example.
  11. Optimization Problem.
  12. Optimization Algorithms
  13. Relaxation.
  14. Rounding.
  15. Time Analysis
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\eta \in (0, 1)$, $\varepsilon > 0$. Denote $B \doteq \frac{1}{\varepsilon} \ln ( \max \left\{ { \frac{2d}{\eta(e^\varepsilon + 1)}, \frac{8 \, e^\varepsilon}{e^{2\varepsilon} - 1} } \right\} )$, and assume that $n \ge B$. There is an algorithm that, given a private version $\tilde{h}$ of a $d$ where $\tilde{f} [t] = \frac{1}{d} | \{\ell \in \mathcal{D} : \tilde{h}[\ell] = t \} |,

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: ${\varepsilon}$-Differentially Privacy DR14
  • Definition 3.1: Circulant Matrix
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Theorem 4.5
  • Lemma 4.6
  • ...and 26 more