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Approximately Optimal Multi-Stream Quickest Change Detection for Gaussian Streams

Joshua Kartzman, Calvin Hawkins, Matthew Hale

TL;DR

This work tackles multi-stream quickest change detection under a one-sample-per-step constraint for Gaussian streams with an unknown post-change mean. It introduces Decaying-$\varepsilon$-FOCuS, a bandit-inspired method that blends a decaying exploration rule with an efficient GLR-based detector to identify a change in mean without discretizing the parameter space. The authors establish ARL guarantees and asymptotic bounds on detection delay that match the Lai-Lorden surrogate, and they demonstrate robustness and scalability through extensive simulations. The approach extends bandit-based change detection to continuous post-change parameters and two-sided shifts, offering practical, provable guarantees for complex sensing environments.

Abstract

This paper considers the bandit quickest change detection problem in which one stream contains a change-point that shifts its distribution by an unknown amount in an unknown direction. We consider an agent that can observe only a single stream at each time, and the goal of the agent is to detect this change as quickly as possible while controlling for false alarms. We propose an algorithm that combines a decaying-$ε$-greedy stream switching rule with an efficient change-point detection algorithm for unknown post-change means. We provide bounds on the expected detection delay and average run length to false alarm for our algorithm, and based on these results we prove our algorithm is approximately optimal with respect to a commonly used surrogate. This work is the first to provide provable guarantees in this setting without strong assumptions such as a discretized post-change parameter set or a lower bound on the magnitude of change.

Approximately Optimal Multi-Stream Quickest Change Detection for Gaussian Streams

TL;DR

This work tackles multi-stream quickest change detection under a one-sample-per-step constraint for Gaussian streams with an unknown post-change mean. It introduces Decaying--FOCuS, a bandit-inspired method that blends a decaying exploration rule with an efficient GLR-based detector to identify a change in mean without discretizing the parameter space. The authors establish ARL guarantees and asymptotic bounds on detection delay that match the Lai-Lorden surrogate, and they demonstrate robustness and scalability through extensive simulations. The approach extends bandit-based change detection to continuous post-change parameters and two-sided shifts, offering practical, provable guarantees for complex sensing environments.

Abstract

This paper considers the bandit quickest change detection problem in which one stream contains a change-point that shifts its distribution by an unknown amount in an unknown direction. We consider an agent that can observe only a single stream at each time, and the goal of the agent is to detect this change as quickly as possible while controlling for false alarms. We propose an algorithm that combines a decaying--greedy stream switching rule with an efficient change-point detection algorithm for unknown post-change means. We provide bounds on the expected detection delay and average run length to false alarm for our algorithm, and based on these results we prove our algorithm is approximately optimal with respect to a commonly used surrogate. This work is the first to provide provable guarantees in this setting without strong assumptions such as a discretized post-change parameter set or a lower bound on the magnitude of change.
Paper Structure (20 sections, 11 theorems, 114 equations, 1 figure, 5 tables, 1 algorithm)

This paper contains 20 sections, 11 theorems, 114 equations, 1 figure, 5 tables, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM FORMULATION AND BACKGROUND
  3. Problem Formulation
  4. CUSUM Procedure
  5. Generalized Likelihood Ratio Procedure
  6. Optimality Results
  7. THE DECAYING-$\epsilon$-FOCUS ALGORITHM
  8. PERFORMANCE GUARANTEES
  9. Average Run Length Constraint
  10. Optimal Detection Delays
  11. Proof Sketches
  12. NUMERICAL EXPERIMENTS
  13. Expected Detection Delay
  14. Average Run Length
  15. CONCLUSION
  16. ...and 5 more sections

Key Result

Theorem 1

Consider an agent using Decaying-$\epsilon$-FOCuS on $M$ streams, where all streams are distributed according to $f_0$, i.e., no change has occurred. Then given a threshold $\lambda>0$, the expected time until a false detection $\tau$ is bounded via as $\lambda\to\infty$, where $g(x)$ is defined as

Figures (1)

  • Figure 1: Expected detection delays for $\mu_1\in[-2,2]$ and $M\in[10,1000].$

Theorems & Definitions (20)

  • Theorem 1
  • Proposition 1
  • Corollary 1
  • Proposition 2
  • Theorem 2
  • Corollary 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 10 more