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Private measures, random walks, and synthetic data

March Boedihardjo, Thomas Strohmer, Roman Vershynin

TL;DR

A polynomial-time algorithm is developed that creates a private measure from a data set that allows us to efficiently construct private synthetic data that are accurate for a wide range of statistical analysis tools.

Abstract

Differential privacy is a mathematical concept that provides an information-theoretic security guarantee. While differential privacy has emerged as a de facto standard for guaranteeing privacy in data sharing, the known mechanisms to achieve it come with some serious limitations. Utility guarantees are usually provided only for a fixed, a priori specified set of queries. Moreover, there are no utility guarantees for more complex - but very common - machine learning tasks such as clustering or classification. In this paper we overcome some of these limitations. Working with metric privacy, a powerful generalization of differential privacy, we develop a polynomial-time algorithm that creates a private measure from a data set. This private measure allows us to efficiently construct private synthetic data that are accurate for a wide range of statistical analysis tools. Moreover, we prove an asymptotically sharp min-max result for private measures and synthetic data for general compact metric spaces. A key ingredient in our construction is a new superregular random walk, whose joint distribution of steps is as regular as that of independent random variables, yet which deviates from the origin logarithmicaly slowly.

Private measures, random walks, and synthetic data

TL;DR

A polynomial-time algorithm is developed that creates a private measure from a data set that allows us to efficiently construct private synthetic data that are accurate for a wide range of statistical analysis tools.

Abstract

Differential privacy is a mathematical concept that provides an information-theoretic security guarantee. While differential privacy has emerged as a de facto standard for guaranteeing privacy in data sharing, the known mechanisms to achieve it come with some serious limitations. Utility guarantees are usually provided only for a fixed, a priori specified set of queries. Moreover, there are no utility guarantees for more complex - but very common - machine learning tasks such as clustering or classification. In this paper we overcome some of these limitations. Working with metric privacy, a powerful generalization of differential privacy, we develop a polynomial-time algorithm that creates a private measure from a data set. This private measure allows us to efficiently construct private synthetic data that are accurate for a wide range of statistical analysis tools. Moreover, we prove an asymptotically sharp min-max result for private measures and synthetic data for general compact metric spaces. A key ingredient in our construction is a new superregular random walk, whose joint distribution of steps is as regular as that of independent random variables, yet which deviates from the origin logarithmicaly slowly.
Paper Structure (47 sections, 29 theorems, 146 equations, 5 figures)

This paper contains 47 sections, 29 theorems, 146 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. A private measure
  4. Uniform accuracy over Lipschitz statistics
  5. A superregular random walk
  6. Comparison to existing work
  7. The architecture of the paper
  8. Background and Notation
  9. Defining metric privacy
  10. Distances between measures
  11. Total variation
  12. Wasserstein distance
  13. A superregular random walk
  14. Heuristics
  15. Formal construction
  16. ...and 32 more sections

Key Result

Lemma 2.3

Let $\mathcal{X}$ be an arbitrary set. Then an algorithm $\mathcal{A}: \mathcal{X}^n \to E$ is $\alpha$-differentially private if an only if $\mathcal{A}$ is $\alpha$-metrically private with respect to the Hamming distance eq: Hamming on $\mathcal{X}^n$.

Figures (5)

  • Figure 1: Faber-Schauder system
  • Figure 2: Haar system
  • Figure 3: The depth-first search tour demonstrates that the TSP of a tree equals twice the sum of lengths of its edges.
  • Figure 4: Chaining: construction of a spanning tree of a metric space.
  • Figure 5: The map $F$ folds an interval $[0,\mathop{\mathrm{TSP}}\nolimits(M)]$ into a Hamiltonian path (a "space-filling curve") of the metric space $T$.

Theorems & Definitions (53)

  • Definition 2.1: Differential Privacy dwork2014algorithmic
  • Definition 2.2: Metric privacy
  • Lemma 2.3: MP vs. DP
  • Theorem 3.1: A superregular random walk
  • Lemma 3.2: Sparsity
  • proof
  • Remark 3.3: Boundedness of paths
  • Proposition 3.4: No boundedness for $\ell^p$-regular potentials
  • proof
  • Lemma 4.1: Private measure yields private synthetic data
  • ...and 43 more