Quickest Change-point Detection in Generalized Linear Models with Variational Estimation

We consider quickest change-point detection in generalized linear models when the post-change parameter is unknown. Such problems arise, for example, when monitoring whether the relationship between a learned representation and an observed response changes over time. Since detection must occur soon after a change, the post-change parameter must be estimated from a limited number of observations, making finite-sample estimation accuracy particularly important. We propose a window-limited CUSUM procedure that estimates the post-change parameter using a variational-inequality (VI) formulation. For noncanonical links, VI estimation can offer favorable finite-sample accuracy over short windows despite the asymptotic efficiency advantage of maximum likelihood estimation. We characterize how estimation error reduces the effective post-change information rate and affects detection delay, and derive finite-window guarantees accounting for both statistical error and incomplete numerical optimization. The resulting procedure retains false-alarm control and achieves first-order asymptotic optimality under suitable conditions. Numerical experiments compare detection delay and update cost for fixed-budget estimation and benchmark VI-WLCUSUM against alternative detectors.