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Privately Counting Partially Ordered Data

Matthew Joseph, Mónica Ribero, Alexander Yu

TL;DR

This work considers differentially private counting when each data point consists of $d$ bits satisfying a partial order and proposes a problem-specific $K$-norm mechanism that runs in time $O(d^2)$.

Abstract

We consider differentially private counting when each data point consists of $d$ bits satisfying a partial order. Our main technical contribution is a problem-specific $K$-norm mechanism that runs in time $O(d^2)$. Experiments show that, depending on the partial order in question, our solution dominates existing pure differentially private mechanisms, and can reduce their error by an order of magnitude or more.

Privately Counting Partially Ordered Data

TL;DR

This work considers differentially private counting when each data point consists of bits satisfying a partial order and proposes a problem-specific -norm mechanism that runs in time .

Abstract

We consider differentially private counting when each data point consists of bits satisfying a partial order. Our main technical contribution is a problem-specific -norm mechanism that runs in time . Experiments show that, depending on the partial order in question, our solution dominates existing pure differentially private mechanisms, and can reduce their error by an order of magnitude or more.

Paper Structure

This paper contains 17 sections, 16 theorems, 8 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Preliminaries
  5. Differential Privacy
  6. Partially Ordered Data
  7. An Efficient Mechanism
  8. Poset Ball
  9. Double Order Polytopes and Double Chain Polytopes
  10. Rejection Sampling the Poset Ball Is Inefficient
  11. Experiments
  12. Choice of Baseline
  13. Random Posets
  14. National Health Interview Survey
  15. Discussion
  16. ...and 2 more sections

Key Result

Lemma 2.2

Given statistic $T$ with $\|\cdot\|$-sensitivity $\Delta$ and database $X$, the $K$-norm mechanism has output density $f_X(y) \propto \exp\left(-\frac{\varepsilon}{\Delta} \cdot \|y - T(X)\|\right)$ and satisfies $\varepsilon$-DP.

Figures (4)

  • Figure 1: $r_{p,d}^2\mathbb{E}_2^2(B_p^d)$ (see \ref{['lem:expected_squared_norm']}).
  • Figure 2: Each point records the average mean squared $\ell_2$ norm ratio between the poset and $\ell_\infty$ balls over 100 trials. A lower value means our mechanism achieves lower error.
  • Figure 3: Each point records the average mean squared $\ell_2$ norm ratio between the poset and $\ell_\infty$ balls over at least 100 trials. We omit extremes of both depth and number of edges, as they do not occur enough times in our overall sample of 5000 random posets to cross the 100-sample threshold.
  • Figure 4: NHIS average mean squared $\ell_2$ norm ratios, from 10,000 trials each.

Theorems & Definitions (38)

  • Definition 2.1: DMNS06
  • Lemma 2.2: HT10
  • Definition 2.3: KN16AS21
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6: HT10
  • Definition 2.7
  • Lemma 3.1
  • proof
  • Definition 3.2
  • ...and 28 more