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High-Probability Bounds For Heterogeneous Local Differential Privacy

Maryam Aliakbarpour, Alireza Fallah, Swaha Roy, Ria Stevens

TL;DR

The paper addresses statistical estimation under local differential privacy with user-specific privacy budgets and high-probability guarantees. It develops tight finite-sample, high-probability bounds for single- and multi-dimensional mean estimation under heterogeneous LDP, and extends to distribution learning via a projection-based, privacy-aware approach; the results include matching minimax lower bounds, establishing near-optimal performance. A key contribution is the weighted-aggregation design that accounts for varying privacy levels, combined with refined high-probability analyses that yield additive log factors in the confidence parameter $\beta$, and novel high-dimensional techniques using hemisphere mixtures. The findings provide principled guidance for mechanism design in heterogeneous privacy settings and show how standard LDP tools can be adapted to achieve robust, high-probability guarantees with practical communication and computation properties.

Abstract

We study statistical estimation under local differential privacy (LDP) when users may hold heterogeneous privacy levels and accuracy must be guaranteed with high probability. Departing from the common in-expectation analyses, and for one-dimensional and multi-dimensional mean estimation problems, we develop finite sample upper bounds in $\ell_2$-norm that hold with probability at least $1-β$. We complement these results with matching minimax lower bounds, establishing the optimality (up to constants) of our guarantees in the heterogeneous LDP regime. We further study distribution learning in $\ell_\infty$-distance, designing an algorithm with high-probability guarantees under heterogeneous privacy demands. Our techniques offer principled guidance for designing mechanisms in settings with user-specific privacy levels.

High-Probability Bounds For Heterogeneous Local Differential Privacy

TL;DR

The paper addresses statistical estimation under local differential privacy with user-specific privacy budgets and high-probability guarantees. It develops tight finite-sample, high-probability bounds for single- and multi-dimensional mean estimation under heterogeneous LDP, and extends to distribution learning via a projection-based, privacy-aware approach; the results include matching minimax lower bounds, establishing near-optimal performance. A key contribution is the weighted-aggregation design that accounts for varying privacy levels, combined with refined high-probability analyses that yield additive log factors in the confidence parameter , and novel high-dimensional techniques using hemisphere mixtures. The findings provide principled guidance for mechanism design in heterogeneous privacy settings and show how standard LDP tools can be adapted to achieve robust, high-probability guarantees with practical communication and computation properties.

Abstract

We study statistical estimation under local differential privacy (LDP) when users may hold heterogeneous privacy levels and accuracy must be guaranteed with high probability. Departing from the common in-expectation analyses, and for one-dimensional and multi-dimensional mean estimation problems, we develop finite sample upper bounds in -norm that hold with probability at least . We complement these results with matching minimax lower bounds, establishing the optimality (up to constants) of our guarantees in the heterogeneous LDP regime. We further study distribution learning in -distance, designing an algorithm with high-probability guarantees under heterogeneous privacy demands. Our techniques offer principled guidance for designing mechanisms in settings with user-specific privacy levels.

Paper Structure

This paper contains 37 sections, 23 theorems, 117 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Single-Dimensional Mean Estimation:
  4. Multi-Dimensional Mean Estimation:
  5. Distribution Learning:
  6. Our Contributions
  7. Single-Dimensional Mean Estimation.
  8. Multi-Dimensional Mean Estimation.
  9. Distribution Learning.
  10. Minimax Quantiles
  11. Notation
  12. Related Works
  13. Single-Dimensional Mean Estimation
  14. Upper Bounds for Bounded Distributions
  15. Upper Bounds for Discrete Distributions
  16. ...and 22 more sections

Key Result

Theorem 1

Let $\boldsymbol{\varepsilon} = \{\varepsilon_i\}_{i=1}^n$ with $\varepsilon_i \leq 1$ for all $i$ and $\beta \in (0, 1)$. Let $\mathcal{P}$ be the family of distributions $P$ such that for any $X \sim P$, $X$ is bounded almost surely. Then, for all $P \in \mathcal{P}$, there exists an $\boldsymbol{

Theorems & Definitions (38)

  • Definition 1: Pure Local Differential Privacy (LDP) Kasiviswanathan2011
  • Theorem 1: Informal version of Theorem \ref{['thm:laplace_mech']} and Theorem \ref{['thm:RR']}
  • Theorem 2: Informal version of Theorem \ref{['thm:highDimUB']}
  • Theorem 3: Informal version of Theorem \ref{['thm:HistEstBassilyUB']}
  • Definition 2: LDP-Minimax Quantiles
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • proof : Proof Sketch
  • ...and 28 more