平均学习动力学中的一阶涨落
Rank-One Fluctuations in Averaging-Learning Dynamics
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中文总结 AI 辅助
研究无外部真实基准的平均学习动力学，通过多布鲁申型收缩等方法，在可和性条件下得出反向乘积收敛到一阶极限，可和扰动下过程收敛到随机共识状态，独立同分布扰动时有中心极限定理，揭示长期涨落渐近一维及相关几何。
中文摘要 AI 辅助
我们研究无外部真实基准的平均学习动力学：参考信号由群体内生生成。该动力学结合时变平均矩阵、学习源矩阵和通常为对角阵的学习矩阵。多布鲁申型收缩控制振荡衰减并产生渐近一致性。在可和性条件下，反向乘积指数收敛到一阶极限。对于可和扰动，过程收敛到随机共识状态。对于独立同分布扰动，我们证明了中心过程的中心极限定理：极限高斯律支持在一致性方向上。因此，尽管是多智能体动力学，长期涨落渐近为一维。我们还记录了一个成对多布鲁申公式，阐明了单类状态下动态一致性类几何。
英文摘要
We study averaging-learning dynamics without an exogenous ground truth: the reference signal is generated endogenously by the population. The dynamics combine a time-varying averaging matrix, a learning-source matrix, and a learning matrix, typically diagonal. Dobrushin-type contraction controls the decay of oscillations and yields asymptotic agreement. Under a summability condition, the backward products converge exponentially to rank-one limits. With summable perturbations, the process converges to a random consensus state. For i.i.d. perturbations, we prove a central limit theorem for the centered process: the limiting Gaussian law is supported on the agreement direction. Thus, despite the multi-agent dynamics, the long-time fluctuations are asymptotically one-dimensional. We also record a pairwise Dobrushin formulation that clarifies the dynamic agreement-class geometry underlying the one-class regime.
