Locally Private Parametric Methods for Change-Point Detection

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.

• The paper develops sharper finite-sample GLRT guarantees for parametric offline change-point detection using martingale concentration, including bounds based on likelihood-ratio sensitivity, KL divergence, and Chernoff information.
• The paper proves that Rényi and infinite-order Jeffreys–Rényi SDPI coefficients are optimized by binary-support input distributions, simplifying privacy-contraction analysis and showing privatized divergence is capped by 2ε under ε-LDP.
• The paper introduces randomized-response and binary-mechanism CPD algorithms, finding regime-dependent superiority and showing that strong privacy reduces the error exponent by roughly tanh²(ε/2), or Θ(ε²) when ε is small.

Problem setting and motivation

This paper studies parametric offline change-point detection (CPD) under local differential privacy (LDP). The setup is the standard non-Bayesian, single-change-point model: a dataset $\mathcal{D} = \{x_1,\dots,x_n\}$ of i.i.d. samples from $P_0$ switches to $P_1$ at an unknown change-point $k^\star \in [n]$, and an analyst must output $\hat{k}$ that is $(\alpha,\beta)$-accurate, meaning $$\mathbb{P}\{\hat{k} \notin [k^\star-\alpha, k^\star+\alpha]\} = \beta$$. In the private variant, each sample passes through an $\varepsilon$-LDP mechanism at the source before reaching the analyst, who receives only the privatized dataset along with the induced distributions $(Q_0,Q_1)$. This removes any reliance on a trusted curator and is motivated by settings such as hospital networks sharing sensitive patient records.

The paper makes three contributions: sharper finite-sample bounds for the non-private GLRT-based estimator; structural results on strong data processing inequality (SDPI) coefficients for Rényi and Jeffreys–Rényi divergences; and two LDP-based CPD algorithms with accuracy guarantees quantifying the cost of privacy.

Improved non-private accuracy guarantees

The core estimator is the generalized log-likelihood ratio test (GLRT), which outputs $$\hat{k} = \arg\max_k \sum_{i=k}^n \log(P_1(x_i)/P_0(x_i))$$. Two quantities govern the analysis: the sensitivity $P_0$0 (the order-$P_0$1 Jeffreys–Rényi divergence, equal to the maximum likelihood-ratio swing between neighboring databases) and $P_0$2.

The main theorem establishes two finite-sample bounds via martingale concentration. Bound A uses maximal Azuma–Hoeffding on bounded-difference martingales constructed from pairwise likelihood-ratio differences, giving

$P_0$3

with $P_0$4. Bound B uses Doob's maximal inequality on an exponential martingale to obtain $P_0$5, where $P_0$6 is the Chernoff information; this bound holds for general alphabets, including continuous ones.

Two implications deserve emphasis. First, both bounds are strictly tighter than the prior result of Zhang et al., which relied on Ottaviani's inequality and was only meaningful for tolerance levels exceeding a threshold involving $P_0$7; the new analysis removes that assumption and improves constants across the range where all bounds are defined. Second, bound A exhibits a counterintuitive behavior: for fixed $P_0$8, increasing the dataset size $P_0$9 does not improve—and can only slightly worsen—the error probability, which saturates at $P_1$0 as $P_1$1. Accuracy is governed by the tolerance level $P_1$2, not by additional post-change data. The authors note empirically that bound B appears tighter than bound A in simulations but do not prove this ordering holds uniformly over all distribution pairs—an honest limitation of the comparison.

SDPI coefficients: binary-support optimality

To analyze privatized algorithms, the paper needs to quantify how an $P_1$3-LDP channel contracts divergences. Its key structural result extends the known binary-optimality of SDPI coefficients for $P_1$4-divergences (Ordentlich and Polyanskiy) to new divergence families:

• For Rényi divergence of every order $P_1$5, the SDPI coefficient $P_1$6 is achieved by input distributions supported on at most two atoms.
• The same holds for the Jeffreys–Rényi divergence of order $P_1$7.

This reduces the coefficient computation from an optimization over the full $P_1$8-dimensional simplex to a search over two-point mixtures. Exploiting this reduction, the paper derives closed forms for $P_1$9-ary symmetric channels: $k^\star \in [n]$0 (achieved by degenerate distributions) and $k^\star \in [n]$1 (achieved by uniform binary distributions). These results have independent applications to statistical estimation lower bounds, data compression, and Markov chain mixing.

A complementary upper bound shows that any $k^\star \in [n]$2-LDP mechanism satisfies $k^\star \in [n]$3 regardless of the inputs, so the privatized separation between pre- and post-change distributions is capped at $k^\star \in [n]$4—a hard ceiling imposed by privacy itself.

Locally private CPD algorithms

Using post-privatization Chernoff information as a performance proxy, the paper identifies regime-dependent optimal mechanisms. It proves that randomized response maximizes $k^\star \in [n]$5 among all $k^\star \in [n]$6-LDP mechanisms for sufficiently large $k^\star \in [n]$7 (low privacy), while the binary mechanism is optimal for sufficiently small $k^\star \in [n]$8 (high privacy), extending Kairouz–Oh–Viswanath's extremal-mechanism results for $k^\star \in [n]$9-divergences to Chernoff information.

Two algorithms follow. RR-CPD privatizes each sample via randomized response over the original alphabet and applies GLRT to the induced distributions. BM-CPD first quantizes the alphabet into two bins via a threshold $\hat{k}$0 (partitioning according to the likelihood ratio $\hat{k}$1), then applies binary randomized response; $\hat{k}$2 is chosen to maximize post-privatization Chernoff information, a choice validated empirically, with $\hat{k}$3 provably as good as $\hat{k}$4 in the high-privacy regime.

Both algorithms admit $\hat{k}$5-accuracy guarantees of the same structural form as the non-private case, with the effective separation attenuated by privacy. For RR-CPD, the Chernoff-type term involves $\hat{k}$6; for BM-CPD it involves $\hat{k}$7, with sensitivity capped at $\hat{k}$8.

Cost of privacy. The unifying quantitative conclusion is that the $\hat{k}$9-LDP constraint multiplies the exponential error exponent by approximately $(\alpha,\beta)$0, i.e., $(\alpha,\beta)$1 up to algorithm- and distribution-dependent constants. Since $(\alpha,\beta)$2 for small $(\alpha,\beta)$3, maintaining fixed accuracy under strong privacy requires inflating the tolerance $(\alpha,\beta)$4 by roughly $(\alpha,\beta)$5. The empirical exponent ratios track this curve closely in Monte Carlo experiments, though the theoretical claim is stated heuristically rather than as a uniform guarantee—a caveat worth noting when interpreting the constant factors.

Empirical validation

Experiments use truncated Poisson distributions ($(\alpha,\beta)$6, $(\alpha,\beta)$7), $(\alpha,\beta)$8, $(\alpha,\beta)$9, with 10,000 Monte Carlo repetitions; Bernoulli, binomial, and truncated geometric variants appear in appendices. The empirical error curves fall below both proposed bounds and well below the prior bound of Zhang et al., confirming the sharpness improvement. At $$\mathbb{P}\{\hat{k} \notin [k^\star-\alpha, k^\star+\alpha]\} = \beta$$0, BM-CPD dominates RR-CPD empirically; sweeping $$\mathbb{P}\{\hat{k} \notin [k^\star-\alpha, k^\star+\alpha]\} = \beta$$1 confirms the predicted crossover—BM-CPD wins in the high-privacy regime ($$\mathbb{P}\{\hat{k} \notin [k^\star-\alpha, k^\star+\alpha]\} = \beta$$2), RR-CPD in the low-privacy regime—with the non-private algorithm uniformly best, as expected.

Limitations and open questions

Several restrictions are explicit. The analysis assumes a single change-point, i.i.d. data within each segment, and finite alphabets for the private results (only bound B extends to continuous alphabets, and only in the non-private setting). The $$\mathbb{P}\{\hat{k} \notin [k^\star-\alpha, k^\star+\alpha]\} = \beta$$3 cost characterization is asymptotic/heuristic rather than a worst-case theorem. The claimed superiority of bound B over bound A is empirical, not proven. The binary mechanism requires knowledge of $$\mathbb{P}\{\hat{k} \notin [k^\star-\alpha, k^\star+\alpha]\} = \beta$$4 to design the quantizer, so its applicability to misspecified models is unaddressed. Open questions raised by the paper include extending these guarantees to online CPD (where detection delay, accuracy, and privacy trade off nontrivially), to non-i.i.d. data, and to multiple change-points under LDP.

Conclusion

The paper strengthens the finite-sample theory of GLRT-based parametric CPD through martingale methods, establishes binary-support optimality of SDPI coefficients for Rényi and Jeffreys–Rényi divergences, and uses these tools to give the first locally private offline CPD algorithms with explicit accuracy guarantees. Its central quantitative finding—that local privacy attenuates the detection error exponent by roughly $$\mathbb{P}\{\hat{k} \notin [k^\star-\alpha, k^\star+\alpha]\} = \beta$$5, scaling as $$\mathbb{P}\{\hat{k} \notin [k^\star-\alpha, k^\star+\alpha]\} = \beta$$6 in the high-privacy regime—provides a clean benchmark for the statistical price of source-level privacy, and the demonstrated regime-dependent superiority of binary versus randomized response mechanisms offers practical guidance for mechanism selection.