We consider a set of agents who are attempting to iteratively learn the 'state of the world' from their neighbors in a social network. Each agent initially receives a noisy observation of the true state of the world. The agents then repeatedly 'vote' and observe the votes of some of their peers, from which they gain more information. The agents' calculations are Bayesian and aim to myopically maximize the expected utility at each iteration.
This model, introduced by Gale and Kariv (2003), is a natural approach to learning on networks. However, it has been criticized, chiefly because the agents' decision rule appears to become computationally intractable as the number of iterations advances. For instance, a dynamic programming approach (part of this work) has running time that is exponentially large in \min(n, (d-1)^t), where n is the number of agents.
We provide a new algorithm to perform the agents' computations on locally tree-like graphs. Our algorithm uses the dynamic cavity method to drastically reduce computational effort. Let d be the maximum degree and t be the iteration number. The computational effort needed per agent is exponential only in O(td) (note that the number of possible information sets of a neighbor at time t is itself exponential in td).
Under appropriate assumptions on the rate of convergence, we deduce that each agent is only required to spend polylogarithmic (in 1/\eps) computational effort to approximately learn the true state of the world with error probability \eps, on regular trees of degree at least five. We provide numerical and other evidence to justify our assumption on convergence rate.
We extend our results in various directions, including loopy graphs. Our results indicate efficiency of iterative Bayesian social learning in a wide range of situations, contrary to widely held beliefs.
朴素社会学习的DeGroot模型假设代理人只交流标量意见。在实践中，代理人不仅传达他们的意见，而且传达他们对这些意见的信心。我们提出了一个模型，捕捉这方面的沟通，将信号的信息量到天真的社会学习的情况。我们提出的模型捕捉了贝叶斯学习和朴素学习的各个方面。在我们的模型中，代理使用贝叶斯规则组合他们邻居的信念，但代理天真地假设他们邻居的信念是独立的。根据初始信念，代理在我们的模型中可能不会达成共识，但我们表明，代理将达成共识下温和的连续性和有界性假设的初始信念。这种最终的共识可以根据每个代理的中心性和信号信息量来明确计算，从而可以精确地理解联合效应。我们将我们的理论应用于新技术的采用。与Banerjee et al. [2018]相反，我们表明，关于新技术的信息最初可以在一个紧密聚集的群体中播种，而不会丢失信息，但前提是代理人可以表达他们的信念。
