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Contraction of Locally Differentially Private Mechanisms

Shahab Asoodeh, Huanyu Zhang

TL;DR

This work examines how locally differentially private mechanisms contract statistical information, deriving tight $\varepsilon$-dependent bounds on $f$-divergences after privatization. It introduces two complementary routes: (i) sharp bounds on contraction coefficients for $\chi^2$, KL, and squared Hellinger, and (ii) a TV-based bound that ensures finite, well-behaved contraction when $\chi^2$ or KL may blow up. These technical results enable locally private versions of the van Trees, Le Cam, Assouad, and mutual-information methods, improving minimax risk analyses for entropy estimation, discrete distribution estimation, nonparametric density estimation, and hypothesis testing. The paper also provides concrete corollaries for private Fisher information, private Le Cam and Assouad bounds, and a flexible private mutual-information approach, with explicit Gaussian-location and hypothesis-testing implications that improve constants and extend to all $\varepsilon \ge 0$.Overall, the framework yields practical, tighter privacy-utility bounds across a range of statistical tasks under LDP, clarifying how privacy budget $\varepsilon$ reshapes fundamental limits and enabling more precise privacy-preserving inference.

Abstract

We investigate the contraction properties of locally differentially private mechanisms. More specifically, we derive tight upper bounds on the divergence between $PK$ and $QK$ output distributions of an $ε$-LDP mechanism $K$ in terms of a divergence between the corresponding input distributions $P$ and $Q$, respectively. Our first main technical result presents a sharp upper bound on the $χ^2$-divergence $χ^2(PK}\|QK)$ in terms of $χ^2(P\|Q)$ and $\varepsilon$. We also show that the same result holds for a large family of divergences, including KL-divergence and squared Hellinger distance. The second main technical result gives an upper bound on $χ^2(PK\|QK)$ in terms of total variation distance $\mathsf{TV}(P, Q)$ and $ε$. We then utilize these bounds to establish locally private versions of the van Trees inequality, Le Cam's, Assouad's, and the mutual information methods, which are powerful tools for bounding minimax estimation risks. These results are shown to lead to better privacy analyses than the state-of-the-arts in several statistical problems such as entropy and discrete distribution estimation, non-parametric density estimation, and hypothesis testing.

Contraction of Locally Differentially Private Mechanisms

TL;DR

This work examines how locally differentially private mechanisms contract statistical information, deriving tight $\varepsilon$-dependent bounds on $f$-divergences after privatization. It introduces two complementary routes: (i) sharp bounds on contraction coefficients for $\chi^2$, KL, and squared Hellinger, and (ii) a TV-based bound that ensures finite, well-behaved contraction when $\chi^2$ or KL may blow up. These technical results enable locally private versions of the van Trees, Le Cam, Assouad, and mutual-information methods, improving minimax risk analyses for entropy estimation, discrete distribution estimation, nonparametric density estimation, and hypothesis testing. The paper also provides concrete corollaries for private Fisher information, private Le Cam and Assouad bounds, and a flexible private mutual-information approach, with explicit Gaussian-location and hypothesis-testing implications that improve constants and extend to all $\varepsilon \ge 0$.Overall, the framework yields practical, tighter privacy-utility bounds across a range of statistical tasks under LDP, clarifying how privacy budget $\varepsilon$ reshapes fundamental limits and enabling more precise privacy-preserving inference.

Abstract

We investigate the contraction properties of locally differentially private mechanisms. More specifically, we derive tight upper bounds on the divergence between and output distributions of an -LDP mechanism in terms of a divergence between the corresponding input distributions and , respectively. Our first main technical result presents a sharp upper bound on the -divergence in terms of and . We also show that the same result holds for a large family of divergences, including KL-divergence and squared Hellinger distance. The second main technical result gives an upper bound on in terms of total variation distance and . We then utilize these bounds to establish locally private versions of the van Trees inequality, Le Cam's, Assouad's, and the mutual information methods, which are powerful tools for bounding minimax estimation risks. These results are shown to lead to better privacy analyses than the state-of-the-arts in several statistical problems such as entropy and discrete distribution estimation, non-parametric density estimation, and hypothesis testing.
Paper Structure (26 sections, 16 theorems, 175 equations, 1 figure, 1 table)

This paper contains 26 sections, 16 theorems, 175 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Additional Related Work
  3. Notation
  4. Preliminaries and Definitions
  5. Main Technical Results
  6. Applications
  7. Locally Private Fisher Information
  8. Private Le Cam's Method: An Improved Version
  9. Private Assouad's Method: An Improved Version
  10. Locally Private Mutual Information Method
  11. Binary Hypothesis Testing under LDP
  12. Tensorization of Contraction Coefficient
  13. Locally Private Cramér-Rao Bound and van Trees Inequality
  14. Missing Proofs
  15. Proof of Theorem \ref{['thm_chiSDPI']}
  16. ...and 11 more sections

Key Result

Theorem 1

If ${\mathsf K}$ is an $\varepsilon$-LDP mechanism, then we have for any $\varepsilon\geq 0$

Figures (1)

  • Figure 1: Contraction coefficient $\eta_{\chi^2}$ for $k$-ary randomized response mechanism with $k=3, 5, 10, 15$. Note that the contraction of binary randomized response coincides with the upper bound given in Theorem \ref{['thm_chiSDPI']}.

Theorems & Definitions (22)

  • Theorem 1
  • Proposition 1: Contraction of $k$-ary randomized response
  • proof
  • Remark 1
  • Theorem 2
  • Lemma 1
  • Corollary 1: Private van Trees Inequality
  • Theorem 3: Improved private Le Cam's method
  • Corollary 2
  • Theorem 4: Improved private Assouad's method
  • ...and 12 more