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Minimax rate for multivariate data under componentwise local differential privacy constraints

Chiara Amorino, Arnaud Gloter

TL;DR

This work analyzes the minimax rates for multivariate data under componentwise local differential privacy (CLDP), where each coordinate is privatized through its own channel with privacy level $α_j$. The authors derive KL-divergence contraction bounds for CLDP, establish minimax lower and upper bounds for nonparametric density estimation and covariance estimation under CLDP, and propose adaptive, data-driven procedures that achieve near-optimal rates (up to logarithmic factors). Key results show that privacy incurs a rate penalty that scales with $n$ and the product of per-component privacy terms, e.g., density estimation under CLDP achieves roughly $(n\prod α_j^2)^{-β/(β+d)}$ (with adaptive variants incurring a $(\log n)^{1+2d}$ factor). The findings quantify the price of privacy in multivariate settings, guide design of privacy mechanisms, and provide practical estimators for CLDP that attain minimax optimality in density and covariance problems.

Abstract

Our research delves into the balance between maintaining privacy and preserving statistical accuracy when dealing with multivariate data that is subject to \textit{componentwise local differential privacy} (CLDP). With CLDP, each component of the private data is made public through a separate privacy channel. This allows for varying levels of privacy protection for different components or for the privatization of each component by different entities, each with their own distinct privacy policies. We develop general techniques for establishing minimax bounds that shed light on the statistical cost of privacy in this context, as a function of the privacy levels $α_1, ... , α_d$ of the $d$ components. We demonstrate the versatility and efficiency of these techniques by presenting various statistical applications. Specifically, we examine nonparametric density and covariance estimation under CLDP, providing upper and lower bounds that match up to constant factors, as well as an associated data-driven adaptive procedure. Furthermore, we quantify the probability of extracting sensitive information from one component by exploiting the fact that, on another component which may be correlated with the first, a smaller degree of privacy protection is guaranteed.

Minimax rate for multivariate data under componentwise local differential privacy constraints

TL;DR

This work analyzes the minimax rates for multivariate data under componentwise local differential privacy (CLDP), where each coordinate is privatized through its own channel with privacy level . The authors derive KL-divergence contraction bounds for CLDP, establish minimax lower and upper bounds for nonparametric density estimation and covariance estimation under CLDP, and propose adaptive, data-driven procedures that achieve near-optimal rates (up to logarithmic factors). Key results show that privacy incurs a rate penalty that scales with and the product of per-component privacy terms, e.g., density estimation under CLDP achieves roughly (with adaptive variants incurring a factor). The findings quantify the price of privacy in multivariate settings, guide design of privacy mechanisms, and provide practical estimators for CLDP that attain minimax optimality in density and covariance problems.

Abstract

Our research delves into the balance between maintaining privacy and preserving statistical accuracy when dealing with multivariate data that is subject to \textit{componentwise local differential privacy} (CLDP). With CLDP, each component of the private data is made public through a separate privacy channel. This allows for varying levels of privacy protection for different components or for the privatization of each component by different entities, each with their own distinct privacy policies. We develop general techniques for establishing minimax bounds that shed light on the statistical cost of privacy in this context, as a function of the privacy levels of the components. We demonstrate the versatility and efficiency of these techniques by presenting various statistical applications. Specifically, we examine nonparametric density and covariance estimation under CLDP, providing upper and lower bounds that match up to constant factors, as well as an associated data-driven adaptive procedure. Furthermore, we quantify the probability of extracting sensitive information from one component by exploiting the fact that, on another component which may be correlated with the first, a smaller degree of privacy protection is guaranteed.
Paper Structure (28 sections, 29 theorems, 239 equations)

This paper contains 28 sections, 29 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Comparison with LDP
  4. Minimax framework
  5. Main results
  6. Bounds on pairwise divergences
  7. Application to privatization of independent sampling
  8. Contraction on $f$-divergence
  9. Applications to statistical inference
  10. Effective privacy level
  11. Locally private joint moment estimation
  12. Local differential private estimator
  13. Application to the covariance estimation
  14. Lower bound for the joint moment estimation
  15. Adpative estimation of the joint moment
  16. ...and 13 more sections

Key Result

Lemma 2.1

If $\bm{Q}$ satisfies the $\bm{\alpha}$-CLDP constraint with $\bm{\alpha}=(\alpha_1,\dots,\alpha_d)$, then ${\color{black} \overline{Q}}$ satisfies ${\color{black} \overline{\alpha}}$-LDP constraint with ${\color{black} \overline{\alpha}}=\sum_{j=1}^d \alpha_j$.

Theorems & Definitions (72)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 1
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Corollary 3.4
  • proof : Proof of Theorem \ref{['th: main bound']}
  • ...and 62 more