Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Growing Alphabets Do Not Automatically Amplify Shuffle Privacy: Obstruction, Estimation Bounds, and Optimal Mechanism Design

Alex Shvets

Abstract

We study neighboring shuffle experiments for epsilon_0-LDP channels along growing alphabets d -> infinity, and optimal mechanism design for frequency estimation under a canonical pairwise chi-squared budget. On the privacy side, we prove an exact compression theorem: the shuffled histogram experiment depends only on the pushforward law of the pairwise likelihood ratio. We establish a sharp universal bound chi^2 <= (e^{epsilon_0}-1)^2/e^{epsilon_0}, construct explicit obstruction families for which the shuffled privacy curve equals binary randomized response for all d, and prove a sharp diluting/persistent dichotomy. On the estimation side, we prove a universal lower bound of order (d-1)/(n chi_*(W)) via Cramer-Rao and Assouad arguments, and show that symmetrization to equivariant channels is WLOG. On the design side, we show calibrated GRR is not optimal. The optimal mechanism is an augmented GRR: fraction p of users applies aggressive GRR with lambda_* = sqrt(d-1), the rest sends a null symbol. This thinning principle is specific to shuffle and has no local-DP counterpart. For low budget 0 < C <= C_*(d), augmented GRR is optimal among all permutation-equivariant channels. GRR is also the unique optimizer within the subset-selection family.

Growing Alphabets Do Not Automatically Amplify Shuffle Privacy: Obstruction, Estimation Bounds, and Optimal Mechanism Design

Abstract

We study neighboring shuffle experiments for epsilon_0-LDP channels along growing alphabets d -> infinity, and optimal mechanism design for frequency estimation under a canonical pairwise chi-squared budget. On the privacy side, we prove an exact compression theorem: the shuffled histogram experiment depends only on the pushforward law of the pairwise likelihood ratio. We establish a sharp universal bound chi^2 <= (e^{epsilon_0}-1)^2/e^{epsilon_0}, construct explicit obstruction families for which the shuffled privacy curve equals binary randomized response for all d, and prove a sharp diluting/persistent dichotomy. On the estimation side, we prove a universal lower bound of order (d-1)/(n chi_*(W)) via Cramer-Rao and Assouad arguments, and show that symmetrization to equivariant channels is WLOG. On the design side, we show calibrated GRR is not optimal. The optimal mechanism is an augmented GRR: fraction p of users applies aggressive GRR with lambda_* = sqrt(d-1), the rest sends a null symbol. This thinning principle is specific to shuffle and has no local-DP counterpart. For low budget 0 < C <= C_*(d), augmented GRR is optimal among all permutation-equivariant channels. GRR is also the unique optimizer within the subset-selection family.
Paper Structure (13 sections, 24 theorems, 249 equations)

This paper contains 13 sections, 24 theorems, 249 equations.

Table of Contents

  1. Introduction
  2. Model and notation
  3. LR-quotient compression
  4. Universal chi-square bound and extremal channels
  5. An explicit obstruction family
  6. A sharp geometric dichotomy
  7. Two poles: GRR versus half-block
  8. Universal lower bound on estimation risk
  9. Symmetrization and reduction to equivariant channels
  10. GRR is not universally optimal. The thinning principle
  11. Low-budget optimality among all permutation-equivariant channels
  12. GRR optimal within the subset-selection family
  13. Discussion

Key Result

Lemma 2.2

Let $W_d$ be $\varepsilon_0$-LDP with $\varepsilon_0<\infty$. For each output symbol $y \in [d]$, either $W_d(y\mid x)=0$ for every input $x$, or $W_d(y\mid x)>0$ for every input $x$.

Theorems & Definitions (59)

  • Definition 2.1
  • Lemma 2.2: Common support under pure LDP
  • proof
  • Lemma 3.1: Exact canonical likelihood ratio
  • proof
  • Theorem 3.2: LR-quotient compression
  • proof
  • Corollary 3.3: The correct invariant
  • proof
  • Lemma 4.1: Bhatia--Davis variance bound
  • ...and 49 more