Locally Private Parametric Methods for Change-Point Detection

TL;DR

The paper addresses parametric change-point detection under local differential privacy, providing a non-private GLRT-based offline CPD with sharpened finite-sample guarantees and two locally private algorithms (randomized response and binary mechanisms). Central to the analysis are strong data processing inequality (SDPI) results for Rényi and Jeffreys-Rényi divergences, which quantify the contraction of information through privacy channels and reveal binary-input optimality. The private CPD methods are evaluated both theoretically and empirically, with bounds showing that privacy degrades the exponential detection-rate by a factor on the order of $\tanh^2(\varepsilon/2)$, and that mechanism choice depends on the privacy regime. The work thus clarifies the cost of local privacy for CPD and provides a principled framework for designing privacy-preserving change-point detectors with concrete performance guarantees.

Abstract

We study parametric change-point detection, where the goal is to identify distributional changes in time series, under local differential privacy. In the non-private setting, we derive improved finite-sample accuracy guarantees for a change-point detection algorithm based on the generalized log-likelihood ratio test, via martingale methods. In the private setting, we propose two locally differentially private algorithms based on randomized response and binary mechanisms, and analyze their theoretical performance. We derive bounds on detection accuracy and validate our results through empirical evaluation. Our results characterize the statistical cost of local differential privacy in change-point detection and show how privacy degrades performance relative to a non-private benchmark. As part of this analysis, we establish a structural result for strong data processing inequalities (SDPI), proving that SDPI coefficients for Rényi divergences and their symmetric variants (Jeffreys-Rényi divergences) are achieved by binary input distributions. These results on SDPI coefficients are also of independent interest, with applications to statistical estimation, data compression, and Markov chain mixing.

Figures (20)

• Figure 1: General framework for parametric change-point detection under local differential privacy. Each data sample in the dataset $\mathcal{D}$ passes through an $\varepsilon-$LDP mechanism at the source. The data analyst performs change-point detection on the privatized dataset $\widetilde{D}$.
• Figure 2: Experiment Plots (see Appendices \ref{['app:npcomp']}, \ref{['app:priv_comp']}, \ref{['app:exp3']} and \ref{['app:cop']} for detailed experiments).
• Figure 3: Good and Bad sub intervals around the true change-point $k^{\star} \in [n]$.
• Figure 4: Comparison of the empirical error probability $\beta$ for Algorithm \ref{['alg:offline_cpd']} and theoretical upper bounds on $\beta$ in the non-private setting, plotted as functions of $\alpha$. We consider binary (Bernoulli) distributions. Left:$P_0 \sim \mathrm{Ber}(0.1)$, $P_1 \sim \mathrm{Ber}(0.4)$. Right:$P_0 \sim \mathrm{Ber}(0.1)$, $P_1 \sim \mathrm{Ber}(0.95)$.
• Figure 5: Comparison of the empirical error probability $\beta$ for Algorithm \ref{['alg:offline_cpd']} and theoretical upper bounds on $\beta$ in the non-private setting, plotted as functions of $\alpha$. We consider truncated Poisson distributions $\mathrm{TPois}(\lambda,m)$ with truncation parameter $m=10$. Left:$P_0 \sim \mathrm{TPois}(1,10)$, $P_1 \sim \mathrm{TPois}(4,10)$. Right:$P_0 \sim \mathrm{TPois}(1,10)$, $P_1 \sim \mathrm{TPois}(10,10)$.

Theorems & Definitions (45)

• Definition 2.1: $(\alpha,\beta)-$accuracy
• Definition 2.2: Chernoff Information
• Definition 2.3: $\rho-$Rényi Divergences
• Definition 2.4: $f-$Divergences
• Definition 2.5: $\varepsilon-$Local Differential Privacy (LDP)
• Theorem 3.1
• proof
• Corollary 3.2
• Remark 3.3
• Definition 4.1: SDPI coefficient