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A Distributionally Robust Optimal Control Approach for Differentially Private Dynamical Systems

Yeongjun Jang, Kaoru Teranishi, Junsoo Kim

Abstract

In this paper, we develop a distributionally robust optimal control approach for differentially private dynamical systems, enabling a plant to securely outsource control computation to an untrusted remote server. We consider a plant that ensures differential privacy of its state trajectory by injecting calibrated noise into its output measurements. Unlike prior works, we assume that the server only has access to an ambiguity set consisting of admissible noise distributions, rather than the exact distribution. To account for this uncertainty, the server formulates a distributionally robust optimal control problem to minimize the worst-case expected cost over all admissible noise distributions. However, the formulated problem is computationally intractable due to the nonconvexity of the ambiguity set. To overcome this, we relax it into a convex Kullback--Leibler divergence ball, so that the reformulated problem admits a tractable closed-form solution.

A Distributionally Robust Optimal Control Approach for Differentially Private Dynamical Systems

Abstract

In this paper, we develop a distributionally robust optimal control approach for differentially private dynamical systems, enabling a plant to securely outsource control computation to an untrusted remote server. We consider a plant that ensures differential privacy of its state trajectory by injecting calibrated noise into its output measurements. Unlike prior works, we assume that the server only has access to an ambiguity set consisting of admissible noise distributions, rather than the exact distribution. To account for this uncertainty, the server formulates a distributionally robust optimal control problem to minimize the worst-case expected cost over all admissible noise distributions. However, the formulated problem is computationally intractable due to the nonconvexity of the ambiguity set. To overcome this, we relax it into a convex Kullback--Leibler divergence ball, so that the reformulated problem admits a tractable closed-form solution.
Paper Structure (10 sections, 33 equations, 3 figures)

This paper contains 10 sections, 33 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Differential privacy of dynamical systems
  4. Problem formulation
  5. Main Results
  6. Tractable reformulation of the optimization problem
  7. Control design
  8. Simulation Results
  9. Conclusion
  10. Proof of Lemma \ref{['lem:mechGauss']}

Figures (3)

  • Figure 1: Plot of $\tau(\eta+W_\tau)$ versus $\tau$, with the optimal value $\tau^*=28.1392$ indicated in red.
  • Figure 2: Performance comparison of the proposed method and standard LQG over 10000 simulations. The mean, $95$th percentile, and worst-case values of the cost $J(\mathcal{K})$ are plotted, while varying the noise parameters $\sigma^2$ and $b$ for (a) the Gaussian and (b) the Laplace mechanisms, respectively.
  • Figure 3: The cost $J(\mathcal{K})$ for different privacy parameters $(\epsilon,\delta)$, averaged over $10000$ simulations. The ratio $\overline{\sigma}^2/\underline{\sigma}^2 = \overline{b}/\underline{b}=1.2$ is fixed for both the (a) Gaussian and (b) Laplace mechanisms.

Theorems & Definitions (4)

  • proof
  • proof
  • proof
  • proof