Slow Convergence of Interacting Kalman Filters in Word-of-Mouth Social Learning
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914969971326976 |
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| author | Krishnamurthy, Vikram Rojas, Cristian |
| author_facet | Krishnamurthy, Vikram Rojas, Cristian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_08447 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Slow Convergence of Interacting Kalman Filters in Word-of-Mouth Social Learning Krishnamurthy, Vikram Rojas, Cristian Machine Learning Theoretical Economics Signal Processing 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. |
| title | Slow Convergence of Interacting Kalman Filters in Word-of-Mouth Social Learning |
| topic | Machine Learning Theoretical Economics Signal Processing |
| url | https://arxiv.org/abs/2410.08447 |