多智能体执行性预测研究提出反射性矩阵，谱半径1为稳定阈值
The Reflexivity Threshold: A Phase Transition for Multi-Agent Performative Prediction
这篇论文提出反射性矩阵 Γ，把多智能体间的相互影响建成网络结构，谱半径跨过 1 就从稳定收敛变成损失线性增长，比以前的标量收缩常数刻画得更细。
arXiv 论文 2610.29577 研究多智能体执行性预测中的反复重训练问题。作者提出反射性矩阵 Γ，记录一个智能体的部署对另一个智能体最优反应的影响强度。当 Γ 的谱半径低于 1 时，反复重训练几何收敛到唯一的执行性稳定均衡，累计超额损失与时间范围无关；谱半径超过 1 时，构造的噪声超临界实例会让所有局部信息学习者的累计超额损失呈线性增长，增长阶为 1/(ρ(Γ)-1)。该阈值由上下界共同刻画，并把先前工作中的单智能体标量阈值作为特例恢复出来。
The Reflexivity Threshold: A Phase Transition for Multi-Agent Performative Prediction
Many learning systems are performative: once deployed, their predictions change the data distribution on which future models are trained. In multi-agent settings, this feedback is routed through a network of interacting learners, so stability depends not only on the strength of performativity but also on who affects whom. We study repeated retraining in multi-agent performative prediction and identify a spectral threshold that separates stable learning from worst-case learning obstruction. Prior work often collapses cross-agent feedback into a scalar contraction constant, missing this network structure. We instead introduce the reflexivity matrix $Γ$, which records how strongly one agent's deployment can move another agent's best response, defined from per-agent gradient sensitivities and curvature parameters. The threshold occurs when the spectral radius of this matrix crosses one. Below the threshold, repeated retraining converges geometrically to a unique performatively stable equilibrium, with cumulative excess loss bounded uniformly in the time horizon. Above the threshold, we construct noisy supercritical instances in which every local-information learner incurs expected cumulative excess loss growing linearly for horizons up to order $1/(ρ(Γ)-1)$. Together, these results characterize the threshold behavior through upper and lower bounds and recover the single-agent scalar threshold from prior work as a special case.
