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PriPL-Tree: Accurate Range Query for Arbitrary Distribution under Local Differential Privacy

Leixia Wang, Qingqing Ye, Haibo Hu, Xiaofeng Meng

TL;DR

PriPL-Tree tackles range queries under Local Differential Privacy by replacing uniform-domain assumptions with a piecewise linear model of the data distribution. The method combines a private PL fitting phase, a PriPL-Tree construction phase, and a refinement phase to produce accurate, low-noise frequency estimates, then extends to multi-dimensional queries using data-aware adaptive grids. The key contributions are the PriPL-Tree itself, which reduces both non-uniform distribution and LDP noise errors, and the adaptive 2-D grid extension with consistency refinement to handle higher dimensions. Extensive experiments on real and synthetic data show substantial accuracy improvements over state-of-the-art methods, especially for non-uniform distributions, validating the practical impact of modeling distributions with PL segments under LDP.

Abstract

Answering range queries in the context of Local Differential Privacy (LDP) is a widely studied problem in Online Analytical Processing (OLAP). Existing LDP solutions all assume a uniform data distribution within each domain partition, which may not align with real-world scenarios where data distribution is varied, resulting in inaccurate estimates. To address this problem, we introduce PriPL-Tree, a novel data structure that combines hierarchical tree structures with piecewise linear (PL) functions to answer range queries for arbitrary distributions. PriPL-Tree precisely models the underlying data distribution with a few line segments, leading to more accurate results for range queries. Furthermore, we extend it to multi-dimensional cases with novel data-aware adaptive grids. These grids leverage the insights from marginal distributions obtained through PriPL-Trees to partition the grids adaptively, adapting the density of underlying distributions. Our extensive experiments on both real and synthetic datasets demonstrate the effectiveness and superiority of PriPL-Tree over state-of-the-art solutions in answering range queries across arbitrary data distributions.

PriPL-Tree: Accurate Range Query for Arbitrary Distribution under Local Differential Privacy

TL;DR

PriPL-Tree tackles range queries under Local Differential Privacy by replacing uniform-domain assumptions with a piecewise linear model of the data distribution. The method combines a private PL fitting phase, a PriPL-Tree construction phase, and a refinement phase to produce accurate, low-noise frequency estimates, then extends to multi-dimensional queries using data-aware adaptive grids. The key contributions are the PriPL-Tree itself, which reduces both non-uniform distribution and LDP noise errors, and the adaptive 2-D grid extension with consistency refinement to handle higher dimensions. Extensive experiments on real and synthetic data show substantial accuracy improvements over state-of-the-art methods, especially for non-uniform distributions, validating the practical impact of modeling distributions with PL segments under LDP.

Abstract

Answering range queries in the context of Local Differential Privacy (LDP) is a widely studied problem in Online Analytical Processing (OLAP). Existing LDP solutions all assume a uniform data distribution within each domain partition, which may not align with real-world scenarios where data distribution is varied, resulting in inaccurate estimates. To address this problem, we introduce PriPL-Tree, a novel data structure that combines hierarchical tree structures with piecewise linear (PL) functions to answer range queries for arbitrary distributions. PriPL-Tree precisely models the underlying data distribution with a few line segments, leading to more accurate results for range queries. Furthermore, we extend it to multi-dimensional cases with novel data-aware adaptive grids. These grids leverage the insights from marginal distributions obtained through PriPL-Trees to partition the grids adaptively, adapting the density of underlying distributions. Our extensive experiments on both real and synthetic datasets demonstrate the effectiveness and superiority of PriPL-Tree over state-of-the-art solutions in answering range queries across arbitrary data distributions.
Paper Structure (51 sections, 3 theorems, 14 equations, 21 figures, 9 tables, 4 algorithms)

This paper contains 51 sections, 3 theorems, 14 equations, 21 figures, 9 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local Differential Privacy (LDP)
  4. Problem Definition
  5. Existing Methods
  6. Observations and Challenges
  7. Private Piecewise Linear Tree
  8. Design Rationale
  9. Workflow of PriPL-Tree
  10. Private PL Fitting
  11. Segment Fitting
  12. Interval Partitioning
  13. PriPL-Tree Construction
  14. User Allocation
  15. Tree Structure Construction
  16. ...and 36 more sections

Key Result

theorem 1

Given a PriPL-Tree with at most $K$ segments (corresponding to $K$ leaf nodes), the error variance of frequencies after weight averaging in refinement (phase 3) is $O\left(\frac{K\cdot \log K}{(1-\alpha) \cdot (K+1) \cdot N \cdot \epsilon^2}\right)$ for non-leaf nodes and $O\left(\frac{\log K}{(1-\a

Figures (21)

  • Figure 1: An Illustration on Non-uniform Errors
  • Figure 2: An Example of PriPL-Tree and HT
  • Figure 3: Workflow of PriPL-Tree
  • Figure 4: An Example of Tree Construction ($N'= N(1-\alpha)$)
  • Figure 5: Examples of Adaptive Grids (Black solid lines represent partitions inherited from PriPL-Trees; blue solid lines indicate newly added partitions; red dashed lines indicate deleted partitions.)
  • ...and 16 more figures

Theorems & Definitions (4)

  • definition 1: $\epsilon$-Local Differential Privacy ($\epsilon$-LDP) duchi2018minimax
  • theorem 1
  • theorem 2
  • lemma 1