Quickest Detection with Social Learning: Interaction of local and global decision makers

We consider how local and global decision policies interact in stopping time problems such as quickest time change detection. Individual agents make myopic local decisions via social learning, that is, each agent records a private observation of a noisy underlying state process, selfishly optimizes its local utility and then broadcasts its local decision. Given these local decisions, how can a global decision maker achieve quickest time change detection when the underlying state changes according to a phase-type distribution? The paper presents four results. First, using Blackwell dominance of measures, it is shown that the optimal cost incurred in social learning based quickest detection is always larger than that of classical quickest detection. Second, it is shown that in general the optimal decision policy for social learning based quickest detection is characterized by multiple thresholds within the space of Bayesian distributions. Third, using lattice programming and stochastic dominance, sufficient conditions are given for the optimal decision policy to consist of a single linear hyperplane, or, more generally, a threshold curve. Estimation of the optimal linear approximation to this threshold curve is formulated as a simulation-based stochastic optimization problem. Finally, the paper shows that in multi-agent sensor management with quickest detection, where each agent views the world according to its prior, the optimal policy has a similar structure to social learning.

💡 Research Summary

The paper investigates the interplay between local and global decision makers in a quickest change‑detection problem when agents learn socially. Each agent observes a noisy signal of an underlying Markovian state that evolves according to a phase‑type distribution. After receiving a private observation, the agent updates its Bayesian belief and then makes a myopic decision that maximizes an immediate utility (typically a trade‑off between false‑alarm and delay costs). This local decision—rather than the raw observation—is broadcast to subsequent agents and to a central decision maker. The central (global) decision maker aggregates the sequence of local actions, updates its own belief, and decides when to stop and declare that a change has occurred.

The authors present four main contributions. First, using Blackwell’s ordering of experiments, they prove that the information loss inherent in social learning makes the optimal expected cost of quickest detection with social learning strictly larger than that of the classical formulation where the global decision maker observes the raw signals. This establishes a fundamental performance penalty for any system that relies on compressed, action‑based communication.

Second, they show that the optimal stopping policy in the belief simplex is generally not a single threshold. Because the belief update after each action is nonlinear and depends on the history of actions, the optimal stopping region can consist of multiple disjoint sub‑regions, i.e., a multi‑threshold structure. This phenomenon is illustrated with numerical examples where the belief space is partitioned into three or more zones, each associated with a different decision.

Third, the paper derives sufficient conditions under which the optimal policy collapses to a much simpler geometric form—a single linear hyperplane (or, more generally, a smooth threshold curve) separating the stopping and continuation regions. The conditions involve monotone likelihood ratio (MLR) ordering of the observation likelihoods, convexity of the cost function, and MLR‑preserving state transition matrices. Under these assumptions, the belief update is monotone, and lattice programming arguments guarantee that the value function is supermodular, leading to a threshold policy.

To make the threshold curve tractable, the authors formulate a simulation‑based stochastic optimization problem that seeks the best linear approximation to the optimal boundary. They employ sample‑average approximation (SAA) together with stochastic gradient descent to estimate the hyperplane parameters that minimize the expected detection cost. Numerical experiments demonstrate that the linear approximation achieves near‑optimal performance, reducing the cost gap to only a few percent relative to the true optimal (non‑linear) policy.

Finally, the authors extend the analysis to a multi‑agent sensor‑management setting where each sensor holds its own prior belief about the state. Even when sensors act independently and only their beliefs (not raw measurements) are communicated to the fusion center, the optimal global policy retains the same single‑hyperplane structure under the same sufficient conditions. This shows that the structural results are robust across different information‑exchange architectures.

Overall, the paper provides a rigorous theoretical framework for quickest detection under social learning, quantifies the inevitable cost of information compression, characterizes the potentially complex shape of optimal policies, and offers practical conditions and algorithms for simplifying those policies. The findings are directly relevant to distributed monitoring applications such as IoT sensor networks, autonomous vehicle fleets, and smart grid fault detection, where communication constraints force agents to share decisions rather than raw data. Future work could explore adversarial settings, non‑phase‑type change dynamics, or reinforcement‑learning approaches to learn the threshold surface online.