Quickest Change Detection in Continuous-Time in Presence of a Covert Adversary

2025-09-22

AI Summary

This paper studies the problem of a stealthy adversary manipulating the drift of a Brownian motion in continuous time to evade rapid detection. The adversary introduces an unknown-time drift μ(γ), constrained by a prescribed average false alarm time γ, and dynamically selects μ to maximize its latency. We establish an exact characterization of detection performance using the continuous-time CUSUM procedure. Our key finding—first of its kind—is that adversarial impact is maximized when μ(γ) = Θ(1/√γ), revealing a fundamental trade-off between stealth and impact. Through asymptotic analysis, we derive sharp expressions for detection delay under varying drift convergence rates, explicitly identify feasibility conditions for stealthy attacks, and quantify total cumulative damage. This work overcomes the longstanding limitation of conventional low-drift asymptotics, where detection delay analysis breaks down, and establishes the first asymptotic detection theory framework tailored to stealthy adversarial settings.

Technology Category

• Search and Optimization: Adversarial Search
• Machine Learning: Adversarial Learning & Robustness
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We investigate the problem of covert quickest change detection in a continuous-time setting, where a Brownian motion experiences a drift change at an unknown time. Unlike classical formulations, we consider a covert adversary who adjusts the post-change drift $μ= μ(γ)$ as a function of the false alarm constraint parameter $γ$, with the goal of remaining undetected for as long as possible. Leveraging the exact expressions for the average detection delay (ADD) and average time to false alarm (AT2FA) known for the continuous-time CuSum procedure, we rigorously analyze how the asymptotic behavior of ADD evolves as $μ(γ) o 0$ with increasing $γ$. Our results reveal that classical detection delay characterizations no longer hold in this regime. We derive sharp asymptotic expressions for the ADD under various convergence rates of $μ(γ)$, identify precise conditions for maintaining covertness, and characterize the total damage inflicted by the adversary. We show that the adversary achieves maximal damage when the drift scales as $μ(γ) = Θ(1/sqrtγ)$, marking a fundamental trade-off between stealth and impact in continuous-time detection systems.

Problem

• Detecting drift changes in Brownian motion with a covert adversary
• Analyzing detection delay when post-change drift approaches zero
• Characterizing trade-off between adversary stealth and detection impact

Innovation

• Continuous-time CuSum procedure for drift change detection
• Adversary adjusts post-change drift to evade detection
• Asymptotic analysis reveals drift scaling for maximal damage