自校准的代价：自适应监控中的精确证据预算与人为制造的盲集
The Price of Self-Calibration: Exact Evidence Budgets and Manufactured Blind Sets in Adaptive Monitoring
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中文总结 AI 辅助
本文精确刻画自校准监控器在任意漂移下的误报率保证代价，提出证据恒等式与盲集理论，证明速度有界类盲集为双倍速度类，并给出更紧的高斯投影功效界。
中文摘要 AI 辅助
自校准监控器在线调整其阈值，以保证在任意漂移下具有规定的长期误报率。我们计算了该保证的代价，并陈述每条定律及其精确的有效域。首先，该保证是一个会计恒等式，与监控器旨在检测的内容无关。两个证据恒等式使在线分位数跟踪器的代价精确：高度为$\delta$的持续阶跃产生的超额报警质量在$\delta/\eta$的一个报警范围内，当$\delta$是增益$\eta$的格点倍数时，路径上恰好为$\delta/\eta$；斜率为$c$的斜坡产生恰好为$c/\eta$的平稳超额率，与累积大小无关，直至边界$c=\eta(1-\alpha)$与报警率上限重合。其次，证书自身的波动遵循精确定律：窗口报警率的标准差为$1/L$阶，而非二项式的$1/\sqrt{L}$，因为窗口质量望远镜式地归结为紧内部状态的差值；闭式常数无需拟合参数即可验证。在二项式尺度上校准的检测器会因$\sqrt{\eta\varphi(q_0)L}$而失准，正确的校准使检测窗口从二次变为反故障速度的线性。第三，任何要求容忍漂移类$\mathcal{D}$的监控器，在任何时间范围、任何规则下，对$\mathcal{D}-\mathcal{D}$中的每个故障都是盲的；证明是刻意初等的两点论证，贡献在于其识别的对象：对于速度有界类，盲集恰好是双倍速度类，跟踪器吸收由其自身增益固定的速度类，因此在宣称吸收漂移为正常的认证制度下，监控器制造了$\mathcal{D}$。一个精确的高斯投影界，比Pinsker界更紧且从不平凡，量化了其外的功效。
英文摘要
Self-calibrating monitors adapt their threshold online to guarantee a prescribed long-run false-alarm rate under arbitrary drift. We compute the price of that guarantee, stating every law with its exact domain of validity. First, the guarantee is an accounting identity, insensitive to what the monitor is meant to detect. Two evidence identities make the cost exact for the online quantile tracker: a persistent step of height $δ$ yields excess alarm mass within one alarm of $δ/η$, and exactly $δ/η$ pathwise when $δ$ is a lattice multiple of the gain $η$; a ramp of slope $c$ yields a stationary excess rate of exactly $c/η$, independent of accumulated size, up to a boundary $c=η(1-α)$ coinciding with the alarm-rate cap. Second, the certificate's own fluctuation obeys an exact law: the windowed alarm rate has standard deviation of order $1/L$, not the binomial $1/\sqrt{L}$, since the windowed mass telescopes to a difference of a tight internal state; the closed-form constant is validated with no fitted parameter. Detectors calibrated on the binomial scale are miscalibrated by $\sqrt{ηφ(q_0)L}$, and correct calibration turns detection windows from quadratic to linear in the inverse fault speed. Third, any monitor required to tolerate a drift class $\mathcal{D}$ is blind, at any horizon and for any rule, to every fault in $\mathcal{D}-\mathcal{D}$; the proof is a deliberately elementary two-point argument and the contribution is the object it identifies: for speed-bounded classes the blind set is exactly the doubled-speed class, and the tracker absorbs a speed class fixed by its own gain, so that under a certification regime declaring absorbed drift normal, the monitor manufactures $\mathcal{D}$. An exact Gaussian projection bound, sharper than Pinsker and never vacuous, quantifies power outside it.
发表机构
- Institut National des Postes et Télécommunications（国家邮电学院）
- Togo AI Lab（多哥人工智能实验室）
机构由 AI 辅助整理，请以论文原文为准。
