Stopping Self-Excitement

From management of service interactions to studies of neural signal propagation, there is a subtle but fundamental challenge that comes with such endogenously spurred activity: when should you say it’s over? Stop too early and you may miss further observations, but stop too late and you’ve wasted too much time. In this paper, we study an optimal stopping problem that aims to balance this omission-excess tradeoff within a cluster of history-driven events. We analyze and prescribe stopping policies for the self-exciting stochastic process through a transformation to the workload process for an M/D/1 queueing system. Though it is intractable to solve the true stopping problem, we prove that our proposed policy admits a uniformly bounded optimality gap. Then, to connect these model-based insights to practice, we draw upon ideas from the algorithms-with-predictions literature and adapt the model-based policies to incorporate contextual information, such as the texts within a service interaction. Here, we propose modified policies that incorporate this outside information to achieve performance consistent with the best of the contextual predictions yet robust enough to inherit strong guarantees from the original, prediction-less policies. We demonstrate the performance of our model and policies on public data of interpersonal communication and AI usage.