Differentially Private Recursive Least Squares Estimation for ARX Systems with Multi-Participants

TL;DR

This work addresses privacy in MP-ARX system identification by introducing a differentially private recursive least-squares (DP-RLS) algorithm. Privacy is achieved by adding Laplacian perturbations to each participant’s data before transmission and performing online RLS updates with perturbed regressors, under conditions of asymptotic stability. The paper establishes $\varepsilon$-DP guarantees for each participant, derives an error bound under a general (weak) excitation condition without requiring independence or boundedness of regressors, and formulates an optimization to select noise intensities that balance privacy with estimation accuracy. Numerical examples illustrate the privacy-utility tradeoffs and the impact of input informativeness on convergence. The results provide a practical, privacy-preserving online identification framework for MP-ARX systems, highlighting the inevitable privacy-utility tradeoff and laying groundwork for future unbiased DP methods in this setting.

Abstract

This paper proposes a differentially private recursive least squares algorithm to estimate the parameter of autoregressive systems with exogenous inputs and multi-participants (MP-ARX systems) and protect each participant's sensitive information from potential attackers. We first give a rigorous differential privacy analysis of the algorithm, and establish the quantitative relationship between the added noises and the privacy-preserving level when the system is asymptotically stable. The asymptotic stability of the system is necessary for ensuring the differential privacy of the algorithm. We then give an estimation error analysis of the algorithm under the general and possible weakest excitation condition without requiring the boundedness, independence and stationarity on the regression vectors. Particularly, when there is no regression term in the system output and the differential privacy only on the system output is considered, $\varepsilon$-differential privacy and almost sure convergence of the algorithm can be established simultaneously. To minimize the estimation error of the algorithm with $\varepsilon$-differential privacy, the existence of the noise intensity is proved. Finally, two examples are given to show the efficiency of the algorithm.

Figures (5)

• Figure 1: Architecture of the problem: multiple participants (i.e. $\mathcal{P}_0$, $\cdots$, $\mathcal{P}_m$ ) collaborate to complete a parameter identification problem that cannot be completed by any individual participant, where participants are low-resource parties with sensitive information that they outsource to a powerful data center (may be one participant). The data center has to solve an identification problem on the sensitive information of the participants
• Figure 2: Estimation error of the algorithm under different inputs
• Figure 3: Estimation error of the algorithm under different privacy indexes when $p=\xi_{1,k}=\xi_{2,k} = 0$
• Figure 4: Estimation error of the algorithm under different variances of the added Laplacian noise
• Figure 5: Estimation error of the algorithm under different $\varepsilon$

Theorems & Definitions (39)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Definition 1: $\delta$-adjacency
• Definition 2: $\varepsilon$-differential privacy
• Remark 6
• Remark 7
• Remark 8