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This paper gives a theoretical account of when repeated retraining stays stable in multi-agent settings where each deployed model shifts the data future models train on. It replaces the usual single scalar contraction constant with a "reflexivity matrix" and shows that the spectral radius crossing one separates convergence to a stable equilibrium from linear excess loss. The results are proven bounds rather than empirical benchmarks, so their practical value depends on how well the per-agent sensitivity and curvature parameters can be estimated in real systems.
The Reflexivity Threshold: A Phase Transition for Multi-Agent Performative Prediction
arXiv:2610.29577v1 Announce Type: new Abstract: 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 $\Gamma$, 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 $1/(\rho(\Gamma)-1)$. 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.
Abstract: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 $\Gamma$, 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 $1/(\rho(\Gamma)-1)$. 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.
Accepted at NeurIPS 2026. 20 pages, including appendix and checklist. Code: this https URL
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2610.29577 [cs.LG]
(or arXiv:2610.29577v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.29577 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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