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Strict Optimality of Frequency Estimation Under Local Differential Privacy

Mingen Pan

Abstract

This paper establishes the strict optimality in precision for frequency estimation under local differential privacy (LDP). We prove that a frequency estimator with a symmetric and extremal configuration, and a constant support size equal to an optimized value, is sufficient to achieve maximum precision. Furthermore, we derive that the communication cost of such an optimal estimator can be as low as $\log_2(\frac{d(d-1)}{2}+1)$, where $d$ denotes the dictionary size, and propose an algorithm to generate this optimal estimator. In addition, we introduce a modified Count-Mean Sketch and demonstrate that it is practically indistinguishable from theoretical optimality with a sufficiently large dictionary size (e.g., $d=100$ for a privacy factor of $ε= 1$). We compare existing methods with our proposed optimal estimator to provide selection guidelines for practical deployment. Finally, the performance of these estimators is evaluated experimentally, showing that the empirical results are consistent with our theoretical derivations.

Strict Optimality of Frequency Estimation Under Local Differential Privacy

Abstract

This paper establishes the strict optimality in precision for frequency estimation under local differential privacy (LDP). We prove that a frequency estimator with a symmetric and extremal configuration, and a constant support size equal to an optimized value, is sufficient to achieve maximum precision. Furthermore, we derive that the communication cost of such an optimal estimator can be as low as , where denotes the dictionary size, and propose an algorithm to generate this optimal estimator. In addition, we introduce a modified Count-Mean Sketch and demonstrate that it is practically indistinguishable from theoretical optimality with a sufficiently large dictionary size (e.g., for a privacy factor of ). We compare existing methods with our proposed optimal estimator to provide selection guidelines for practical deployment. Finally, the performance of these estimators is evaluated experimentally, showing that the empirical results are consistent with our theoretical derivations.
Paper Structure (32 sections, 30 theorems, 91 equations, 1 figure, 1 table)

This paper contains 32 sections, 30 theorems, 91 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Background
  4. Local Differential Privacy (LDP)
  5. Frequency Estimation
  6. Extremal Configuration
  7. Symmetric Configuration
  8. Strict Optimality in Precision
  9. Communication Cost
  10. Optimal Algorithms
  11. Subset Selection
  12. Optimized Count Mean Sketch
  13. Weighted Subset Selection
  14. Choosing an Optimal Algorithm
  15. Experiment
  16. ...and 17 more sections

Key Result

Proposition 1

Denote $d$, $\epsilon$, $n$, and $\hat{f}$ as dictionary size (number of all possible inputs), privacy factor, dataset size, and frequency estimator, respectively. When $d \ge e^{\epsilon} + 1 ,$ we have Otherwise,

Figures (1)

  • Figure 1: $\mathcal{L}_1$ and $\mathcal{L}_2$ losses vs. privacy factor $\epsilon$ given the Zipf dataset. See Section \ref{['sec:zipf']} for details.

Theorems & Definitions (31)

  • Proposition 1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.1
  • Theorem 2.4
  • Theorem 2.5
  • Claim 2.1
  • Lemma 2.2
  • Theorem 2.6
  • ...and 21 more