多代理社会学习中信息传递越多次，收敛越慢，重加权可恢复最优速度。
Slow Convergence of Interacting Kalman Filters in Word-of-Mouth Social Learning
- 多级卡尔曼滤波逐级传递均值估计，形成链式学习机制。
- 当代理数为 $m$ 时，协方差以 $k^{-(2^m-1)}$ 速率下降，远慢于标准 $k^{-1}$。
- 人为调整先验权重可恢复 $k^{-1}$ 最优收敛率，适合研究信息传播效率者。
我们研究了 $m$ 个依次运行的卡尔曼滤波代理构成的口口相传社会学习模型。首个滤波器接收原始观测，后续每个滤波器接收前一滤波器条件均值的含噪测量。第 $m$ 个滤波器更新先验。当 $m=2$ 且观测为高斯随机变量的含噪测量时，协方差以 $k^{-1/3}$ 速率趋于零，而非标准卡尔曼滤波的 $O(k^{-1})$。本文证明：对 $m$ 个代理，协方差收敛速率为 $k^{-(2^m-1)}$，即学习速度随代理数指数级下降。我们还表明，通过人为调整每步先验权重，可使学习速率恢复为最优的 $k^{-1}$。结果表明，在口口相传学习中，重加权先验可实现最优学习速率。
原文摘要 · Abstract (English)
We consider word-of-mouth social learning involving $m$ Kalman filter agents that operate sequentially. The first Kalman filter receives the raw observations, while each subsequent Kalman filter receives a noisy measurement of the conditional mean of the previous Kalman filter. The prior is updated by the $m$-th Kalman filter. When $m=2$, and the observations are noisy measurements of a Gaussian random variable, the covariance goes to zero as $k^{-1/3}$ for $k$ observations, instead of $O(k^{-1})$ in the standard Kalman filter. In this paper we prove that for $m$ agents, the covariance decreases to zero as $k^{-(2^m-1)}$, i.e, the learning slows down exponentially with the number of agents. We also show that by artificially weighing the prior at each time, the learning rate can be made optimal as $k^{-1}$. The implication is that in word-of-mouth social learning, artificially re-weighing the prior can yield the optimal learning rate.
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