Pollak's Minimax Quickest Change Detection: Non-Asymptotic Optimality

Abstract

Although Pollak's minimax formulation is one of the central frameworks in quickest change detection (QCD), the strongest general optimality results available for it are predominantly \emph{asymptotic}, applying as the average run-length-to-false-alarm constraint, $\gamma$, tends to infinity. Exact results for finite $\gamma$ have previously been available only for special models or restricted regimes. This paper addresses the general finite-$\gamma$ problem over the complete class of randomized, history-dependent stopping rules for i.i.d. change models. The key technical development is a \emph{survival-process representation} that recasts the optimization of Pollak's minimax criterion over stopping times as an equivalent linear variational optimization over admissible survival processes. Although this formulation is infinite-dimensional, it establishes the existence of an optimizer and provides an exact characterization of the finite-$\gamma$ Pollak minimax value. This characterization, in turn, provides a principled basis for constructing computable stopping rules whose performance can be made arbitrarily close to the optimum. The resulting rules are driven by a recursively updated weighted likelihood-ratio statistic with generally time-varying injections and boundaries. Importantly, this structure is not imposed a priori: the optimization is carried out over the full class of randomized, history-dependent stopping rules, and the Shiryaev--Roberts form emerges naturally from the solution. In particular, the classical Shiryaev--Roberts recursion arises as the time-homogeneous special case. Finally, under a likelihood-ratio floor condition, the framework yields closed-form exact minimax solutions for a nontrivial class of change models.