Faster Change Detection Achieved with New Theoretical Limit of Error Rate

Scientists are tackling the fundamental challenge of quickly identifying changes in data streams while maintaining a controlled false alarm rate. Ashwin Ram and Aaditya Ramdas, both from Carnegie Mellon University, alongside Ashwin Ram and Aaditya Ramdas et al., present a novel approach to sequential change detection for bounded means. Their work establishes a universal lower bound on detection delay, representing the theoretical limit of how quickly a change can be identified under specific conditions. Significantly, they demonstrate that this limit is achievable in practice, offering a benchmark for evaluating and designing effective change detection algorithms, and providing a uniform minimax guarantee for separated mean shifts.

Fundamental limits to speed in quickest changepoint detection under distribution uncertainty

Researchers have established a fundamental limit on the speed of detecting changes in data streams, even when the initial and altered states are not fully known. This work addresses the challenge of quickest changepoint detection where the pre-change and post-change distributions belong to broad families, a common scenario in real-world applications like sensor monitoring and anomaly detection.
A key difficulty lies in characterizing the optimal detection delay when the most challenging pre-change distribution depends on the unknown post-change distribution, rendering traditional detection methods ineffective. To overcome this, the study derives a universal lower bound for any changepoint detector calibrated to maintain a specific average run length, a measure of false alarm rate.

This lower bound, of the order log(γ)/KLinf(Q, P), applies in the low type-I error regime, where γ represents the false alarm constraint and KLinf denotes a specific information divergence. The researchers demonstrate that this theoretical limit is achievable by constructing a detector for bounded mean detection settings that matches this lower bound with the same constant factor.

This signifies a significant advancement in the efficiency of change detection algorithms. Furthermore, for scenarios involving shifts in mean values, a uniform minimax guarantee is established, confirming the optimality of the developed detector across a range of possible changes. The research introduces a near least-favorable pre-change law, Pδ, satisfying KL(Q∥Pδ) ≤I(Q; P) + δ, and employs a block argument to ensure a conditional null probability of stopping at most f/γ within each block of data.

This approach allows for the identification of windows where atypical evidence accumulation necessitates a change detection decision. The resulting bounded mean detector, denoted as T BM γ, achieves the derived lower bound, demonstrating its effectiveness in practical applications. This work builds upon existing methods like Page’s CUSUM and Shiryaev-Roberts procedures, extending their applicability to more complex and realistic scenarios with composite distribution families. The findings have implications for various fields, including statistical process control, financial monitoring, and network security, where rapid and reliable change detection is crucial.

Establishing a fundamental limit for quickest detection under composite distribution constraints

A universal lower bound for quickest changepoint detection forms the core of this work, derived under Average Run Length (ARL) constraints where both pre- and post-change distributions are composite. The research addresses a significant challenge in characterizing the minimum detection delay when the most difficult pre-change distribution depends on the unknown post-change distribution.

Conventional likelihood-ratio approaches, typically used for Page-CUSUM and Shiryaev-Roberts procedures, are demonstrably inapplicable in this complex scenario. To establish this lower bound, the study leverages analysis in the low type-I error regime, specifically of order, to provide a general result applicable to any ARL-calibrated detector.

Achievability of this bound was then demonstrated through a matching upper bound, confirmed within the important setting of bounded mean detection. This involved proving that the derived lower bound is not merely theoretical but can be attained in practice, with the same constant factor. Furthermore, for the specific case of separated mean shifts, a uniform minimax guarantee of achievability was established across a range of alternative distributions.

This guarantee reinforces the robustness and general applicability of the proposed methodology. The research meticulously details the mathematical framework used to derive these bounds and demonstrate their attainability, contributing to a more complete understanding of optimal sequential change detection strategies.

Fundamental limits on detection delay via Kullback-Leibler divergence and average run length

Researchers established a universal lower bound of 1 divided by the Kullback-Leibler divergence, I(Q; P), for any Average Run Length (ARL)-calibrated changepoint detector in the low type-I error regime. This bound applies to any composite pre-change and post-change law classes, P and Q, respectively, and holds under the constraint that the expected stopping time, Tγ, satisfies infimum over P of the expected value of Tγ under the pre-change law P∞ is greater than or equal to γ.

The work demonstrates achievability of this lower bound by proving a matching upper bound in the bounded mean detection setting, with the same constant factor. For separated mean shifts, a uniform minimax guarantee of this achievability was also derived over the alternative distributions. Specifically, the limit inferior of the cumulative detection delay, CQ(Tγ), as γ approaches infinity, is greater than or equal to 1 divided by I(Q; P), where I(Q; P) represents the Kullback-Leibler information between the post-change law Q and the pre-change law P.

This result establishes a fundamental limit on the performance of any changepoint detection scheme operating under an ARL constraint. Corollary 2 extends this to a minimax statement, indicating that the limit inferior of the supremum over Q in Q of I(Q; P) multiplied by CQ(Tγ) divided by log γ is greater than or equal to 1, assuming Q contains at least one distribution with a finite Kullback-Leibler divergence from P.

The research introduces a series of lemmas and statements that underpin these bounds, focusing on the information gap between pre- and post-change distributions. These findings are particularly relevant to sequential change detection problems where the pre- and post-change laws are complex and unknown.

Optimal detection delay via Kullback-Leibler divergence projection and Shiryaev-Roberts criteria

In composite changepoint detection, characterizing the optimal detection delay presents a significant challenge when the most difficult pre-change distribution depends on the unknown post-change distribution. This work establishes a universal lower bound for any Average Run Length (ARL)-calibrated changepoint detector operating in a low type-I error regime, demonstrating a limit of order .

This bound is achieved through the development of a tight matching upper bound within the bounded mean detection setting, confirming the optimality of the proposed approach. Furthermore, a uniform minimax guarantee of this achievability is derived for separated mean shifts, indicating robustness across a range of alternative distributions.

The key information quantity driving the first-order change detection asymptotics under an ARL constraint is identified as the Kullback-Leibler divergence projection. A mixture Shiryaev-Roberts stopping rule, based on one-step betting factors, is constructed to demonstrate this tightness. Limitations acknowledged by the authors include the current focus on bounded mean detection, and the analysis is presently limited to independent and identically distributed data.

Future research directions involve extending these results beyond the bounded mean model and exploring interpolation between the current sharp constant analysis and existing non-asymptotic, non-i.i.d. detection frameworks. Such advancements may yield computationally efficient and constant-optimal stopping rules that are both distribution-free under complex null hypotheses and leverage additional structural assumptions, such as independence, to further enhance performance.