Sequential Change Detection with Differential Privacy

TL;DR

A differentially private (DP) variant of the CUSUM procedure is introduced that injects calibrated Laplace noise into both the vanilla CUSUM statistics and the detection threshold, preserving the recursive simplicity of the classical CUSUM statistics while ensuring per-sample differential privacy.

Abstract

Sequential change detection is a fundamental problem in statistics and signal processing, with the CUSUM procedure widely used to achieve minimax detection delay under a prescribed false-alarm rate when pre- and post-change distributions are fully known. However, releasing CUSUM statistics and the corresponding stopping time directly can compromise individual data privacy. We therefore introduce a differentially private (DP) variant, called DP-CUSUM, that injects calibrated Laplace noise into both the vanilla CUSUM statistics and the detection threshold, preserving the recursive simplicity of the classical CUSUM statistics while ensuring per-sample differential privacy. We derive closed-form bounds on the average run length to false alarm and on the worst-case average detection delay, explicitly characterizing the trade-off among privacy level, false-alarm rate, and detection efficiency. Our theoretical results imply that under a weak privacy constraint, our proposed DP-CUSUM procedure achieves the same first-order asymptotic optimality as the classical, non-private CUSUM procedure. Numerical simulations are conducted to demonstrate the detection efficiency of our proposed DP-CUSUM under different privacy constraints, and the results are consistent with our theoretical findings.

Figures (5)

• Figure 1: Heatmaps of the effective privacy factor $h(\epsilon, A_\delta)$ under a Gaussian mean shift from $N(0,1)$ to $N(\mu,1)$, for $\mu$ in $\{0.1, 0.25, 0.5\}$. Each panel shows how $h$ varies with $\epsilon \in (0, 3)$ and $\delta \in (0, 1)$. The dashed red curve denotes $\epsilon = 2A_\delta$, i.e., $h(\epsilon, A_\delta) = 1$; above this curve, the DP-CUSUM procedure is proved to be asymptotically optimal by Eq. \ref{['eq:wadd-asym2']}.
• Figure 2: Average detection delay versus average run length of the DP-CUSUM procedure under Laplace distributions at various privacy levels $\epsilon$ for: (a) mean shift from 0 to 0.2; (b) mean shift from 0 to 0.5. The average run length and detection delay are averaged over 10,000 trials.
• Figure 3: Comparison of average detection delay of the DP-CUSUM procedure and the baseline method OnlinePCPD cummings2018differentially under Laplace distributions with different privacy levels for: (a) mean shift from 0 to 0.2; (b) mean shift from 0 to 0.5. The average run length and detection delay are averaged over 10,000 trials. Window size in OnlinePCPD is set as 700 according to cummings2018differentially.
• Figure 4: Average detection delay versus average run length of the DP-CUSUM procedure under Normal distributions at various privacy levels for: (a) mean shift from 0 to 0.1; (b) mean shift from 0 to 0.5. The average run length and detection delay are averaged over 10,000 trials.
• Figure 5: Comparison of average detection delay of the DP-CUSUM procedure and the baseline method cummings2018differentially under Normal distributions with different privacy levels for: (a) mean shift from 0 to 0.1; (b) mean shift from 0 to 0.5. The average run length and detection delay are averaged over 10,000 trials. Window size in OnlinePCPD is set as 700 according to cummings2018differentially.

Theorems & Definitions (27)

• Lemma 1: Performance of exact CUSUM xu2021optimum
• Definition 1: $\epsilon$-DP
• Definition 2: $\epsilon$-DP Sequential Detection Procedure
• Definition 3: Sensitivity of $\ell$
• Theorem 1: DP Guarantee
• Theorem 2: ARL of DP-CUSUM
• Lemma 2: Worst-Case Average Detection Delay
• Theorem 3: WADD of DP-CUSUM
• Remark 1: Relaxed DP for $n>n_0$
• Theorem 4: ($\epsilon,\delta$)-DP guarantee