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Design of Stochastic Quantizers for Privacy Preservation

Le Liu, Yu Kawano, Ming Cao

TL;DR

This work investigates how stochastic quantizers can simultaneously preserve privacy and enable reliable tracking in networked control systems. It first establishes a finite-horizon $(0,\delta)$-differential privacy condition for static stochastic quantizers and derives an upper bound on tracking error, revealing a fundamental privacy-utility trade-off governed by the quantization step $d$. To improve this trade-off, the authors introduce dynamic stochastic quantizers (with time-varying step $d(k)$) that can achieve stronger privacy without sacrificing control when the system matrix is Schur stable, and they extend the approach to unstable systems by adding input Gaussian noise. Simulations on a motion-control example corroborate that dynamic quantizers offer superior privacy and control performance relative to static ones. These results provide design guidelines for privacy-aware quantization in networked control, including strategies for unstable plants via noise augmentation.

Abstract

In this paper, we examine the role of stochastic quantizers for privacy preservation. We first employ a static stochastic quantizer and investigate its corresponding privacy-preserving properties. Specifically, we demonstrate that a sufficiently large quantization step guarantees $(0, δ)$ differential privacy. Additionally, the degradation of control performance caused by quantization is evaluated as the tracking error of output regulation. These two analyses characterize the trade-off between privacy and control performance, determined by the quantization step. This insight enables us to use quantization intentionally as a means to achieve the seemingly conflicting two goals of maintaining control performance and preserving privacy at the same time; towards this end, we further investigate a dynamic stochastic quantizer. Under a stability assumption, the dynamic stochastic quantizer can enhance privacy, more than the static one, while achieving the same control performance. We further handle the unstable case by additionally applying input Gaussian noise.

Design of Stochastic Quantizers for Privacy Preservation

TL;DR

This work investigates how stochastic quantizers can simultaneously preserve privacy and enable reliable tracking in networked control systems. It first establishes a finite-horizon -differential privacy condition for static stochastic quantizers and derives an upper bound on tracking error, revealing a fundamental privacy-utility trade-off governed by the quantization step . To improve this trade-off, the authors introduce dynamic stochastic quantizers (with time-varying step ) that can achieve stronger privacy without sacrificing control when the system matrix is Schur stable, and they extend the approach to unstable systems by adding input Gaussian noise. Simulations on a motion-control example corroborate that dynamic quantizers offer superior privacy and control performance relative to static ones. These results provide design guidelines for privacy-aware quantization in networked control, including strategies for unstable plants via noise augmentation.

Abstract

In this paper, we examine the role of stochastic quantizers for privacy preservation. We first employ a static stochastic quantizer and investigate its corresponding privacy-preserving properties. Specifically, we demonstrate that a sufficiently large quantization step guarantees differential privacy. Additionally, the degradation of control performance caused by quantization is evaluated as the tracking error of output regulation. These two analyses characterize the trade-off between privacy and control performance, determined by the quantization step. This insight enables us to use quantization intentionally as a means to achieve the seemingly conflicting two goals of maintaining control performance and preserving privacy at the same time; towards this end, we further investigate a dynamic stochastic quantizer. Under a stability assumption, the dynamic stochastic quantizer can enhance privacy, more than the static one, while achieving the same control performance. We further handle the unstable case by additionally applying input Gaussian noise.
Paper Structure (16 sections, 9 theorems, 84 equations, 3 figures)

This paper contains 16 sections, 9 theorems, 84 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Formulation
  4. A Motivating Example for Employing Stochastic Quantziers
  5. Trade-off Analysis for Static Stochastic Quantizers
  6. Differential Privacy
  7. Privacy Analysis
  8. Trade-off between Control and Privacy Performances
  9. Improving Trade-off by Dynamic Stochastic Quantizers
  10. Privacy Protection for Unstable Systems
  11. Simulations
  12. Conclusions
  13. Proof of Theorem \ref{['thm:dp']}
  14. Proof of Theorem \ref{['thm:J']}
  15. Proof of Theorem \ref{['thm:dyn_quantizer_arbi']}
  16. ...and 1 more sections

Key Result

Theorem III.3

Given $k \in {\mathbb Z}_{+}$, $\zeta > 0$, and $\delta \in (0, 1)$, a mechanism eq:mech with a static stochastic quantizer static is $(0,\delta)$ differentially private for $\operatorname{Adj}_1^\zeta$ at $k$ if the quantization step $d$ satisfies

Figures (3)

  • Figure 1: Mechanism diagram: The system sends its quantized information to the fusion center, and then the fusion center returns a control input to the system through communication networks, where local information in the green dash box is not eavesdropped while information in the red dash box can be.
  • Figure 2: Deterministic quantizer versus stochastic quantizer
  • Figure 3: Dynamic versus static stochastic quantizers: tracking output $y_1 = x_1$

Theorems & Definitions (25)

  • Remark II.5
  • Remark II.7
  • Example II.8
  • Definition III.1
  • Definition III.2
  • Theorem III.3
  • proof
  • Corollary III.4
  • proof
  • Theorem III.5
  • ...and 15 more