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Privacy Preservation for Statistical Input in Dynamical Systems

Le Liu, Yu Kawano, Ming Cao

TL;DR

The paper addresses privacy preservation for stochastic inputs in linear dynamical systems by introducing a Wasserstein-based adjacency to capture distribution-level differences between inputs. It develops a Gaussian-mechanism framework that leverages two key lemmas linking symmetrized KL divergence to ${\mathcal{W}}_2$ and bounding outputs under affine mappings, yielding a practical condition on the noise covariance ${\Sigma}_{V_t}$ to ensure $(0,\delta)$-DP for Adj$_2^c$. A specialized result for publicly known initial states reduces conservatism, and a building-automation example demonstrates that occupancy-related privacy can be protected while quantifying the trade-off between privacy (via $\delta$) and utility. The work advances privacy in networked control by treating privacy of probabilistic inputs, not just deterministic data, and provides a concrete noisy-data design strategy for DP in discrete-time LTI systems.

Abstract

This paper addresses the challenge of privacy preservation for statistical inputs in dynamical systems. Motivated by an autonomous building application, we formulate a privacy preservation problem for statistical inputs in linear time-invariant systems. What makes this problem widely applicable is that the inputs, rather than being assumed to be deterministic, follow a probability distribution, inherently embedding privacy-sensitive information that requires protection. This formulation also presents a technical challenge as conventional differential privacy mechanisms are not directly applicable. Through rigorous analysis, we develop strategy to achieve $(0, δ)$ differential privacy through adding noise. Finally, the effectiveness of our methods is demonstrated by revisiting the autonomous building application.

Privacy Preservation for Statistical Input in Dynamical Systems

TL;DR

The paper addresses privacy preservation for stochastic inputs in linear dynamical systems by introducing a Wasserstein-based adjacency to capture distribution-level differences between inputs. It develops a Gaussian-mechanism framework that leverages two key lemmas linking symmetrized KL divergence to and bounding outputs under affine mappings, yielding a practical condition on the noise covariance to ensure -DP for Adj. A specialized result for publicly known initial states reduces conservatism, and a building-automation example demonstrates that occupancy-related privacy can be protected while quantifying the trade-off between privacy (via ) and utility. The work advances privacy in networked control by treating privacy of probabilistic inputs, not just deterministic data, and provides a concrete noisy-data design strategy for DP in discrete-time LTI systems.

Abstract

This paper addresses the challenge of privacy preservation for statistical inputs in dynamical systems. Motivated by an autonomous building application, we formulate a privacy preservation problem for statistical inputs in linear time-invariant systems. What makes this problem widely applicable is that the inputs, rather than being assumed to be deterministic, follow a probability distribution, inherently embedding privacy-sensitive information that requires protection. This formulation also presents a technical challenge as conventional differential privacy mechanisms are not directly applicable. Through rigorous analysis, we develop strategy to achieve differential privacy through adding noise. Finally, the effectiveness of our methods is demonstrated by revisiting the autonomous building application.

Paper Structure

This paper contains 15 sections, 6 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Contribution
  4. Organization
  5. Notation
  6. Problem Formulation
  7. A Motivating Example
  8. System Description
  9. Definition of Differential Privacy
  10. Differential Privacy with Output Noise
  11. Numerical Experiments
  12. Conclusion
  13. Proof of Lemma \ref{['lem:disinq']}
  14. Proof of Lemma \ref{['lem:aff']}
  15. Proof of Theorem \ref{['thm:dp']}

Key Result

Lemma III.1

For two Gaussian distributions ${\mathbb P}_1 = {\mathcal{N}}_{n}(m_1, \Sigma_1)$ and ${\mathbb P}_2 = {\mathcal{N}}_{n}(m_2, \Sigma_2)$, the symmetrized KL divergence and the Wasserstein distance between ${\mathbb P}_1$ and ${\mathbb P}_2$ satisfy the following inequality, where ${\mathrm {KL}}( {\mathbb P}_1 \| {\mathbb P}_2)$ denotes the KL divergence between ${\mathbb P}_1$ and ${\mathbb P

Figures (2)

  • Figure 1: Output Performance with $\delta = 0.1$
  • Figure 2: Output Performance with $\delta = 0.2$

Theorems & Definitions (16)

  • Definition II.1
  • Remark II.2
  • Definition II.3
  • Lemma III.1
  • proof
  • Lemma III.2
  • proof
  • Theorem III.3
  • proof
  • Corollary III.4
  • ...and 6 more