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Improved Lower Bound for Differentially Private Facility Location

Pasin Manurangsi

TL;DR

A lower bound of $\tilde{\Omega}\left(\min\left\{\log n, \sqrt{\log n}{\epsilon}}\right\}\right)$ is given on the expected approximation ratio of any $\epsilon-DP algorithm, which is the first evidence that the approximation ratio has to grow with the size of the metric space.

Abstract

We consider the differentially private (DP) facility location problem in the so called super-set output setting proposed by Gupta et al. [SODA 2010]. The current best known expected approximation ratio for an $ε$-DP algorithm is $O\left(\frac{\log n}{\sqrtε}\right)$ due to Cohen-Addad et al. [AISTATS 2022] where $n$ denote the size of the metric space, meanwhile the best known lower bound is $Ω(1/\sqrtε)$ [NeurIPS 2019]. In this short note, we give a lower bound of $\tildeΩ\left(\min\left\{\log n, \sqrt{\frac{\log n}ε}\right\}\right)$ on the expected approximation ratio of any $ε$-DP algorithm, which is the first evidence that the approximation ratio has to grow with the size of the metric space.

Improved Lower Bound for Differentially Private Facility Location

TL;DR

A lower bound of is given on the expected approximation ratio of any $\epsilon-DP algorithm, which is the first evidence that the approximation ratio has to grow with the size of the metric space.

Abstract

We consider the differentially private (DP) facility location problem in the so called super-set output setting proposed by Gupta et al. [SODA 2010]. The current best known expected approximation ratio for an -DP algorithm is due to Cohen-Addad et al. [AISTATS 2022] where denote the size of the metric space, meanwhile the best known lower bound is [NeurIPS 2019]. In this short note, we give a lower bound of on the expected approximation ratio of any -DP algorithm, which is the first evidence that the approximation ratio has to grow with the size of the metric space.
Paper Structure (6 sections, 4 theorems, 14 equations)

This paper contains 6 sections, 4 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Proof Overview.
  4. Preliminaries
  5. Main Proof
  6. Open Questions

Key Result

Theorem 4

For any $0 < \epsilon \leq O(1)$ and any sufficiently large $n$, there exists a metric space $\mathcal{M}$ (with $n$ points) such that any $\epsilon$-DP algorithm for SOFL on $\mathcal{M}$ in the super-set output setting must incur an expected approximation ratio of at least $\Omega\left(\min\left\{

Theorems & Definitions (10)

  • Definition 1: Adjacent Datasets
  • Definition 2: Differential Privacy DworkMNS06
  • Definition 3: Super-set Output Facility Location (SOFL)
  • Theorem 4
  • Theorem 5: erdos1963regulare
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • proof : Proof of \ref{['thm:lb-general']}