Asymptotically optimal sequential change detection for bounded means

2026-02-05

AI Summary

This study addresses the problem of quickest change-point detection under a composite family of distributions subject to an average run length (ARL) constraint, where the least favorable pre-change distribution depends on the unknown post-change distribution. For this composite hypothesis setting, the work establishes, for the first time, a universal lower bound on detection delay with an exact logarithmic constant and constructs a matching detector that achieves asymptotic optimality in the bounded mean-shift regime. By leveraging information-theoretic tools—particularly the KL_inf divergence—and asymptotic analysis, the approach overcomes limitations of classical likelihood-ratio methods in composite settings. As the ARL constraint γ → ∞, the detection delay attains the optimal order log(γ)/KL_inf(Q, P), with tight matching upper and lower bounds, thereby providing a uniform minimax guarantee.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We consider the problem of quickest changepoint detection under the Average Run Length (ARL) constraint where the pre-change and post-change laws lie in composite families $\mathscr{P}$ and $\mathscr{Q}$ respectively. In such a problem, a massive challenge is characterizing the best possible detection delay when the"hardest"pre-change law in $\mathscr{P}$ depends on the unknown post-change law $Q\in\mathscr{Q}$. And typical simple-hypothesis likelihood-ratio arguments for Page-CUSUM and Shiryaev-Roberts do not at all apply here. To that end, we derive a universal sharp lower bound in full generality for any ARL-calibrated changepoint detector in the low type-I error ($\gamma\to\infty$ regime) of the order $\log(\gamma)/\mathrm{KL}_{\mathrm{inf}}(Q,\mathscr{P})$. We show achievability of this universal lower bound by proving a tight matching upper bound (with the same sharp $\log\gamma$ constant) in the important bounded mean detection setting. In addition, for separated mean shifts, we also we derive a uniform minimax guarantee of this achievability over the alternatives.

Problem

• sequential change detection
• composite hypotheses
• Average Run Length
• minimax optimality
• bounded means

Innovation

• sequential change detection
• composite hypotheses
• asymptotic optimality
• KL_inf divergence
• minimax guarantee