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Slow Convergence of Interacting Kalman Filters in Word-of-Mouth Social Learning

Vikram Krishnamurthy, Cristian Rojas

TL;DR

It is proved that for word-of-mouth social learning involving Kalman filter agents, the covariance decreases to zero as $k^{-(2^{m}-1)}$, i.e, the learning slows down exponentially with the number of agents.

Abstract

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.

Slow Convergence of Interacting Kalman Filters in Word-of-Mouth Social Learning

TL;DR

It is proved that for word-of-mouth social learning involving Kalman filter agents, the covariance decreases to zero as , i.e, the learning slows down exponentially with the number of agents.

Abstract

We consider word-of-mouth social learning involving 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 -th Kalman filter. When , and the observations are noisy measurements of a Gaussian random variable, the covariance goes to zero as for observations, instead of in the standard Kalman filter. In this paper we prove that for agents, the covariance decreases to zero as , 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 . The implication is that in word-of-mouth social learning, artificially re-weighing the prior can yield the optimal learning rate.

Paper Structure

This paper contains 4 sections, 4 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction and Problem Formulation
  2. Problem Formulation
  3. Analysis
  4. Recovering the optimal convergence rate

Key Result

Theorem 1

In the word-of-mouth social learning problem, the public belief has a precision $\rho_k^{(m)}$ satisfying where $f\colon \mathbb{R}_+ \to \mathbb{R}_+$ is strictly increasing and such that there exists an $M > 0$ for which $\lambda_e \leqslant f(x) \leqslant M x$ for all $x > 0$. Furthermore,

Figures (1)

  • Figure 1: Three Interacting Kalman Filters

Theorems & Definitions (13)

  • Theorem 1
  • Remark 1
  • proof
  • Theorem 2
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 3
  • ...and 3 more