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Quick Change Detection in Discrete-Time in Presence of a Covert Adversary

Amir Reza Ramtin, Philippe Nain, Don Towsley

TL;DR

This work addresses covert quickest change detection in discrete time when the post-change distribution depends on the false-alarm constraint and approaches the pre-change distribution as the constraint grows. It develops an asymptotic CuSum framework with $q_\gamma\to q$, establishes overshoot-vanishing conditions, and derives exact ADD and AT2FA scalings in terms of the Kullback–Leibler divergences, revealing the regimes that enable covertness where $n(\gamma)=\Theta(\gamma)$. A key contribution is the precise asymptotics for $n(\gamma)$, including a Lambert-W-based function $G(y)$ that captures the transition between detection and covertness, and rigorous treatment of the discrete-time overshoot. The theory is then specialized to Gaussian and exponential observation models, yielding explicit covertness conditions and ADD scaling, with practical implications for adversarial settings in security-sensitive applications. Overall, the results provide exact scaling laws and design insights for covert quickest change detection and quantify how rapidly the post-change distribution must converge to the pre-change distribution to preserve covertness.

Abstract

We study the problem of covert quickest change detection in a discrete-time setting, where a sequence of observations undergoes a distributional change at an unknown time. Unlike classical formulations, we consider a covert adversary who has knowledge of the detector's false alarm constraint parameter $γ$ and selects a stationary post-change distribution that depends on it, seeking to remain undetected for as long as possible. Building on the theoretical foundations of the CuSum procedure, we rigorously characterize the asymptotic behavior of the average detection delay (ADD) and the average time to false alarm (AT2FA) when the post-change distribution converges to the pre-change distribution as $γ\to \infty$. Our analysis establishes exact asymptotic expressions for these quantities, extending and refining classical results that no longer hold in this regime. We identify the critical scaling laws governing covert behavior and derive explicit conditions under which an adversary can maintain covertness, defined by ADD = $Θ(γ)$, whereas in the classical setting, ADD grows only as $\mathcal{O}(\log γ)$. In particular, for Gaussian and Exponential models under adversarial perturbations of their respective parameters, we asymptotically characterize ADD as a function of the Kullback--Leibler divergence between the pre- and post-change distributions and $γ$.

Quick Change Detection in Discrete-Time in Presence of a Covert Adversary

TL;DR

This work addresses covert quickest change detection in discrete time when the post-change distribution depends on the false-alarm constraint and approaches the pre-change distribution as the constraint grows. It develops an asymptotic CuSum framework with , establishes overshoot-vanishing conditions, and derives exact ADD and AT2FA scalings in terms of the Kullback–Leibler divergences, revealing the regimes that enable covertness where . A key contribution is the precise asymptotics for , including a Lambert-W-based function that captures the transition between detection and covertness, and rigorous treatment of the discrete-time overshoot. The theory is then specialized to Gaussian and exponential observation models, yielding explicit covertness conditions and ADD scaling, with practical implications for adversarial settings in security-sensitive applications. Overall, the results provide exact scaling laws and design insights for covert quickest change detection and quantify how rapidly the post-change distribution must converge to the pre-change distribution to preserve covertness.

Abstract

We study the problem of covert quickest change detection in a discrete-time setting, where a sequence of observations undergoes a distributional change at an unknown time. Unlike classical formulations, we consider a covert adversary who has knowledge of the detector's false alarm constraint parameter and selects a stationary post-change distribution that depends on it, seeking to remain undetected for as long as possible. Building on the theoretical foundations of the CuSum procedure, we rigorously characterize the asymptotic behavior of the average detection delay (ADD) and the average time to false alarm (AT2FA) when the post-change distribution converges to the pre-change distribution as . Our analysis establishes exact asymptotic expressions for these quantities, extending and refining classical results that no longer hold in this regime. We identify the critical scaling laws governing covert behavior and derive explicit conditions under which an adversary can maintain covertness, defined by ADD = , whereas in the classical setting, ADD grows only as . In particular, for Gaussian and Exponential models under adversarial perturbations of their respective parameters, we asymptotically characterize ADD as a function of the Kullback--Leibler divergence between the pre- and post-change distributions and .
Paper Structure (34 sections, 21 theorems, 223 equations, 1 figure)

This paper contains 34 sections, 21 theorems, 223 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic elements of change-point detection in discrete-time
  3. Limiting behavior of $n(\gamma)$ when $q_1$ depends on $\gamma$
  4. Application to Gaussian and exponential models
  5. Gaussian probability density functions
  6. Exponential probability density functions
  7. Conclusion
  8. Proof of Proposition \ref{['prop:asymptotic-gaussian']}
  9. Proof that (\ref{['lem:sufficient1']}) holds for Gaussian pdfs
  10. Case when $\sigma\equiv 0$
  11. Case when $\sigma\not\equiv 0$
  12. Case when $-\infty < \liminf_\gamma \tau \leq \limsup_\gamma\tau <\infty$
  13. Case when $\lim_{\gamma} \tau=\pm \infty$
  14. Proof that (\ref{['lem:sufficient2']}) holds for Gaussian pdfs
  15. Case when $\sigma\equiv 0$
  16. ...and 19 more sections

Key Result

Proposition 3.1

Assume that conditions (result005)-(result006) are satisfied. Then, for $a<0<b$, with $A=e^a$ and $B=e^b$.

Figures (1)

  • Figure 1: $v\to H(\sqrt{u_2})$ for $v\in [0,15]$

Theorems & Definitions (25)

  • Proposition 3.1
  • Lemma 3.1
  • Proposition 3.2
  • Lemma 3.2
  • Remark 3.1: Sufficient condition for (\ref{['result005']})-(\ref{['result006']}) to hold
  • Proposition 3.3: Limiting behavior of $n(\gamma)$
  • Corollary 3.1
  • Remark 3.2: Asymptotic behavior of $h^\star(\gamma)$
  • Proposition 3.4
  • Proposition 4.1: Gaussian pdfs
  • ...and 15 more