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Pure Differential Privacy for Functional Summaries via a Laplace-like Process

Haotian Lin, Matthew Reimherr

TL;DR

This work introduces a novel mechanism for DP functional summary release: the Independent Component Laplace Process (ICLP) mechanism, which treats the summaries of interest as truly infinite-dimensional objects, thereby addressing several limitations of existing mechanisms.

Abstract

Many existing mechanisms to achieve differential privacy (DP) on infinite-dimensional functional summaries often involve embedding these summaries into finite-dimensional subspaces and applying traditional DP techniques. Such mechanisms generally treat each dimension uniformly and struggle with complex, structured summaries. This work introduces a novel mechanism for DP functional summary release: the Independent Component Laplace Process (ICLP) mechanism. This mechanism treats the summaries of interest as truly infinite-dimensional objects, thereby addressing several limitations of existing mechanisms. We establish the feasibility of the proposed mechanism in multiple function spaces. Several statistical estimation problems are considered, and we demonstrate one can enhance the utility of sanitized summaries by oversmoothing their non-private counterpart. Numerical experiments on synthetic and real datasets demonstrate the efficacy of the proposed mechanism.

Pure Differential Privacy for Functional Summaries via a Laplace-like Process

TL;DR

This work introduces a novel mechanism for DP functional summary release: the Independent Component Laplace Process (ICLP) mechanism, which treats the summaries of interest as truly infinite-dimensional objects, thereby addressing several limitations of existing mechanisms.

Abstract

Many existing mechanisms to achieve differential privacy (DP) on infinite-dimensional functional summaries often involve embedding these summaries into finite-dimensional subspaces and applying traditional DP techniques. Such mechanisms generally treat each dimension uniformly and struggle with complex, structured summaries. This work introduces a novel mechanism for DP functional summary release: the Independent Component Laplace Process (ICLP) mechanism. This mechanism treats the summaries of interest as truly infinite-dimensional objects, thereby addressing several limitations of existing mechanisms. We establish the feasibility of the proposed mechanism in multiple function spaces. Several statistical estimation problems are considered, and we demonstrate one can enhance the utility of sanitized summaries by oversmoothing their non-private counterpart. Numerical experiments on synthetic and real datasets demonstrate the efficacy of the proposed mechanism.
Paper Structure (55 sections, 11 theorems, 114 equations, 10 figures, 3 tables, 1 algorithm)

This paper contains 55 sections, 11 theorems, 114 equations, 10 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution:
  3. Related Work:
  4. Preliminaries
  5. Notations.
  6. Functional Summaries and Spaces
  7. Differential Privacy
  8. Finite-Dimensional Representation-wise Laplace
  9. The ICLP Mechanism and its Feasibility
  10. Independent Component Laplace Process
  11. Feasibility in Separable Hilbert Spaces
  12. Example of $\mathcal{H}_{C}$ and $\mathcal{H}_{1,C}$ for Smooth Functions
  13. Extensions To Space of Continuous Functions
  14. Methodology
  15. Generalized Obtainment of Qualified Summaries
  16. ...and 40 more sections

Key Result

Theorem 1

Let $C$ represent the covariance operator of the ICLP, with $\{\lambda_j\}_{j\geq 1}$ and $\{\phi_j\}_{j\geq 1}$ denoting its eigenvalues and eigenfunctions, respectively. If $C$ belongs to the trace class, i.e. its non-negative decreasing real eigenvalues $\{\lambda_j\}_{j\geq 1}$ satisfy $\sum_{j\

Figures (10)

  • Figure 1: MSE for PSS and PCV approaches to select regularization parameter $\psi$ under different mechanisms, sample size, and true mean functions. The values reported are averages over $100$ independent replicated experiments. Both axes are in the base 10 log scale.
  • Figure 2: MSE for different mechanisms under different sample sizes $n$ and true mean functions. The values reported are averages over $100$ independent replicated experiments. Both axes are in the base 10 log scale.
  • Figure 3: Non-private (Blue) and 10 random realization of private (Red) KDE curves. (a) ICLP mechanism with $\eta = 1.25$ (b) ICLP mechanism with $\eta = 1.5$ (c) FRL and (d)-(f) Bernstein mechanism with $K=10,30,50$.
  • Figure 4: 3D plot of non-private (Blue) and private (Red) KDE over $\mathbb{R}^{2}$. (a)-(c) ICLP mechanism with $\eta = 1.01, 1.05, 1.2$ (d) FRL and (e)-(f) Bernstein mechanism with $K=5,10$.
  • Figure 5: Non-private sample mean and private means for different mechanisms with Matérn kernel $C_{\frac{3}{2}}$ and $C_{\frac{5}{2}}$ with $\epsilon = 1$. The curves in light grey indicate the original samples.
  • ...and 5 more figures

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem 1
  • Theorem 2: Equivalence of ICLP measure
  • Lemma 1: Density of ICLP
  • Theorem 3: Impossibility of The ICLP Mechanism
  • Theorem 4: The ICLP Mechanism
  • Theorem 5: Feasiability in Space of Continuous Functions
  • Theorem 6: Global Sensitivity Analysis
  • ...and 17 more