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Finite-Horizon Quickest Change Detection Balancing Latency with False Alarm Probability

Yu-Han Huang, Venugopal V. Veeravalli

TL;DR

The paper studies a finite-horizon quickest change detection problem where false alarms are limited by $P_\infty(\tau \le T) \le \delta_F$ and latency is the smallest $\ell$ with $P_\nu(\tau \ge \nu+\ell) \le \delta_D$ for all feasible change-points. A universal $\Omega(\log T)$ latency lower bound is derived, and detectors are constructed that are order-optimal in $T$: TVT-CuSum and TVT-SR for known $f_0,f_1$ with time-varying thresholds, and GLR/GSR for unknown but sub-Gaussian densities. The detectors control false alarms and achieve latency $O(\log T)$, verified by simulations that also show gaps between bounds and practical performance. The results enable robust change detection in piecewise stationary and non-stationary learning contexts, with explicit guidance on horizon scaling and detection strategies.

Abstract

A finite-horizon variant of the quickest change detection (QCD) problem that is of relevance to learning in non-stationary environments is studied. The metric characterizing false alarms is the probability of a false alarm occurring before the horizon ends. The metric that characterizes the delay is \emph{latency}, which is the smallest value such that the probability that detection delay exceeds this value is upper bounded to a predetermined latency level. The objective is to minimize the latency (at a given latency level), while maintaining a low false alarm probability. Under the pre-specified latency and false alarm levels, a universal lower bound on the latency, which any change detection procedure needs to satisfy, is derived. Change detectors are then developed, which are order-optimal in terms of the horizon. The case where the pre- and post-change distributions are known is considered first, and then the results are generalized to the non-parametric case when they are unknown except that they are sub-Gaussian with different means. Simulations are provided to validate the theoretical results.

Finite-Horizon Quickest Change Detection Balancing Latency with False Alarm Probability

TL;DR

The paper studies a finite-horizon quickest change detection problem where false alarms are limited by and latency is the smallest with for all feasible change-points. A universal latency lower bound is derived, and detectors are constructed that are order-optimal in : TVT-CuSum and TVT-SR for known with time-varying thresholds, and GLR/GSR for unknown but sub-Gaussian densities. The detectors control false alarms and achieve latency , verified by simulations that also show gaps between bounds and practical performance. The results enable robust change detection in piecewise stationary and non-stationary learning contexts, with explicit guidance on horizon scaling and detection strategies.

Abstract

A finite-horizon variant of the quickest change detection (QCD) problem that is of relevance to learning in non-stationary environments is studied. The metric characterizing false alarms is the probability of a false alarm occurring before the horizon ends. The metric that characterizes the delay is \emph{latency}, which is the smallest value such that the probability that detection delay exceeds this value is upper bounded to a predetermined latency level. The objective is to minimize the latency (at a given latency level), while maintaining a low false alarm probability. Under the pre-specified latency and false alarm levels, a universal lower bound on the latency, which any change detection procedure needs to satisfy, is derived. Change detectors are then developed, which are order-optimal in terms of the horizon. The case where the pre- and post-change distributions are known is considered first, and then the results are generalized to the non-parametric case when they are unknown except that they are sub-Gaussian with different means. Simulations are provided to validate the theoretical results.

Paper Structure

This paper contains 12 sections, 10 theorems, 62 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Universal Lower Bound on Latency
  4. Candidate Change Detectors
  5. Known Pre- and Post-Change Distributions
  6. Unknown Pre- and Post-Change Distributions
  7. Experimental Results
  8. Conclusions
  9. Proof of Theorem \ref{['thm:latency_pre-post-known']}
  10. Proof of Theorem \ref{['thm:low_bound_latency']}
  11. Proof of Lemma \ref{['lem:GLR-kl-unknown-pre-post']}
  12. Proof of Theorem \ref{['thm:latency_pre-post-unknown']}

Key Result

Theorem 1

For all $\delta_{\mathrm{F}}, \delta_{\mathrm{D}} \in \left( 0, 1 \right)$ such that $\delta_{\mathrm{F}} + \delta_{\mathrm{D}}<1$ as $T \to \infty$.

Figures (1)

  • Figure 1: Latencies of TVT-CuSum, TVT-SR, and GLR tests with fixed $\delta_{\mathrm{F}} = \delta_{\mathrm{D}} = 0.01$ and varying $T \in \left\{ 5000, 10000, 20000, 50000, 100000 \right\}$.

Theorems & Definitions (12)

  • Theorem 1: Asymptotic Lower Bound on Latency
  • Theorem 2: TVT-CuSum and TVT-SR tests
  • Remark 1: Decreasing false alarm and latency levels
  • Lemma 1
  • Theorem 3: GLR and GSR tests
  • Corollary 1
  • Remark 2
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 2 more